In "splendid isolation" While browsing the Net for some articles related to the history of the Whittaker-Shannon sampling theorem, so important to our digital world today, I came across this passage by H. D. Luke in The Origins of the Sampling Theorem:
However, this history also reveals a process which is often apparent in theoretical problems in technology or physics: first the practicians put forward a rule of thumb, then the theoreticians develop the general solution, and finally someone discovers that the mathematicians have long since solved the mathematical problem which it contains, but in "splendid isolation."
Other interesting examples?
(Matrices and Bohr's Quantum Mechanics of course. Someone could elaborate on the sampling theorem if they wish.)
 A: Cormack and Hounsfield received the 1979 Nobel prize in medicine for their work on CT scans. Cormack, a physicist, published his mathematical work on this in 1963, to essentially no response. Hounsfield, an engineer, built the first CT scanner in 1971 unaware of Cormack's work.  Cormark included the following in his Nobel prize speech: "If a fine beam of gamma-rays of intensity $I_0$ is incident on the body and the emerging intensity is $I$, then the
measurable quantity is $g = \ln(I_0/I) = \int_L f ds$, where $f$ is
the variable absorption coefficient along the line $L$.
Hence if $f$ is a function in two dimensions, and $g$ is
known for all lines [...], the question is: Can $f$ be
determined if $g$ is known? This seemed like a problem
which would have been solved before, probably in the
19th century, but a literature search and enquiries of
mathematicians provided no information about it.
Fourteen years would elapse before I learned that Radon
had solved this problem in 1917."
Fourteen years after Cormack's work means 1977, so Radon's work was rediscovered by the people involved with creating CT scan technology only after CT scan's had been around for several years.  (Search on "Radon transform" for more information.)
Radon's work was rediscovered multiple times:

*

*Cramer and Wold (1936) in probability theory,


*Ambartsumian (1936) in astronomy,


*Bracewell (1956) in astronomy,


*De Rosier and Klug (1968) in chemistry.
In fact, Radon's basic idea was worked out before Radon, by Funk (1916) and Lorentz (1905). This work of Lorentz was unpublished, but a formula he found is mentioned in a paper by Bockwinkel in 1906. More on this history is in Cormack's survey paper
Computed tomography: some history and recent developments, pp. 35--42 in "Computed tomography: Proceedings of Symposia in Applied Mathematics" 27, AMS, 1983.
Shortly before the work of Cormack, Oldendorf (a medical doctor in LA) published a paper in 1961 describing a crude CT scanner he had built out of household parts, such as model railroad tracks (!) but it went unnoticed. Hounsfield acknowledged it, but Oldendorf was not included in the Nobel prize list with Cormack and Hounsfield. He once said in an interview "I think Professor Cormack was selected [for the Nobel prize] because he worked out all the line integrals mathematically. [...] I didn't provide any mathematical treatment of it, and that apparently carried a lot of weight with the Nobel committee. See https://en.wikipedia.org/wiki/William_H._Oldendorf for more on his story.
The mathematical and engineering concepts in CT scan technology, with applications
to medical imaging, were worked out in an obscure journal in Kiev by S. T. Tetelbaum
in 1957-58, before Oldendorf!
A: One example that springs to mind are the Dirac equation and Clifford algebras. 
Dirac wanted to take the square root of the Klein-Gordon equation, and calculations showed that he needed 4 "numbers" $\gamma_i$ such that $\gamma_i \gamma_j + \gamma_j \gamma_i = 2\eta_{ij}\text{Id}_4$ with $\eta$ the $4\times 4$ diagonal matrix of the Minkowski metric.
He found 4 complex $4\times 4$ matrices which satisfied these equation. Later physicists found that a general theory of such matrices was given in the 19th century, the theory of Clifford algebras.
A: In 1954 Chen-Ning Yang and Robert Mills discovered nonabelian gauge fields in a physical context (in order to understand the strong force),
only to realize later that the same notion has been discovered in 1950 by Charles Ehresmann in a purely mathematical context.  Related notions, e.g., Cartan connections, has been known to mathematicians for many years before 1950.
A: Quantum mechanics of Born, Heisenberg, and Jordan.
From Physics in my Generation (Springer, 1969) by Max Born:
"In Gottingen we also took part in the attempts to distill the unknown mechanics of the atom out of the experimental results ... The art of guessing correct formulas ... was brought to considerable perfection ...
This period was brought to a sudden end by Heisenberg ... He cut the Gordian knot ... he demanded that the theory should be built up by means of quadratic arrays ... one must find a rule ... for the multiplication of such arrays ...
By consideration of known examples discovered by guesswork, Heisenberg found this rule ...
Heisenberg's rule left me no peace, and after a week of intensive thought and trial, I suddenly remembered an algebraic theory that I had learned from my teacher, Rosanes, in Breslau. Such quadratic arrays are quite familiar to mathematicians, and are called matrices ...
(Born writes down the now iconic [p,q]=pq-qp=iħ.)
My excitement over this result was like that of the mariner who, after long voyaging, sees the land from afar..."
Edit (Mar 2014): In addition, according to Harold Davis in The Theory of Linear Operators (Principia Press, 1936, pg. 199), the commutator [q,p]=1 "was apparently first studied by Charles Graves as early as 1857." Davis goes on to use the commutator to get some "normal ordering" results obtained by Graves and to expand on them.
Edit (Jan 2015) Charles' brother John Graves discovered the octonians (octaves, see Wikipedia) in 1843 and is credited by Hamilton in encouraging his search for the quaternions.
Edit (Jul, 2020) Kwaśniewski cites the relations constructed by Charles Graves
$$[f(a),b] = c f'(a)$$
with $[a,b] = c$ and $[a,c]=[b,c]=0$.
[From "How the work of Gian Carlo Rota had influenced my group research and life" in which Kwasniewski cites O.V. Viskov "On One Result of George Boole" (in Russian), who, in turn, attributes these to Charles Graves in "On the principles which regulate the interchange of symbols in certain symbolic equations," Proc. Royal Irish Academy vol. 6, 1853-1857, pp. 144-15. This pops up in the umbral Sheffer calculus as the Pincherle derivative (circa 1933) with $a=L$, a lowering/destruction/ annihilation and $R=b$, a raising/creation op, or vice versa. Think of the prototypical  $R=x$ and $L=D$ acting on $x^n$. The Pincherle derivative is a delta op, which lowers the degree of polynomials by one. Graves also published a generalized Taylor series shift op which can serve as an umbral substitution, or composition operator in the umbral, Sheffer-Rota finite operator calculus. This all precedes the ladder operators of quantum mechanics by two generations.]
(Edit Oct. 2020) From the biography of Dirac by Helge Kragh via Michael Fowler, Graduate Classical Mechanics:
Dirac made the connection with Poisson brackets on a long Sunday walk, mulling over Heisenberg’s uv vu − (as it was written). He suddenly but dimly remembered what he called “these strange quantities”—the Poisson brackets—which he felt might have properties corresponding to the quantum mathematical formalism Heisenberg was building. But he didn’t have access to advanced dynamics
books until the college library opened the next morning, so he spent a sleepless night. First thing Monday, he read the relevant bit of Whittaker’s Analytical Dynamics, and saw he was correct.
(Interesting that Hamilton was in possession of pretty much the full mathematical apparatus to develop basic quantum mechanics. Of course he had no inkling of quantum phenomena and died when Boltzmann was only 21, so probably did not even suspect the deep role of probability in explaining classical physical phenomena.)
A: When Kepler was trying to work out the orbits of the planets, he wrote something to the effect of, "If only they were ellipses!" as he knew the Greeks had worked that theory out 1500 years earlier. Of course, eventually he convinced himself that they actually were ellipses. Is this the kind of thing you have in mind? 
A: Rooted trees and numerical methods for differential equations.
Excerpt from "What are Butcher series, really? The story of rooted trees and numerical methods for evolution equations" by McLachlan, Modin, Munthe-Kaas, and Verdier:
"Robert Henry ‘Robin’ Merson (1921–1992) was a scientist at the Royal Aircraft Establishment, Farnborough, UK, who was invited along with more senior numerical analysts to a conference on Data Processing and Automatic Computing Machines at Australia’s Weapons Research Establishment in Salisbury, South Australia. It seems like a long way to go for a conference in 1957. However, the UK was still performing above-ground atomic bomb tests in South Australia at that time and the Australian government was very keen to be a part of the emerging era. Merson’s work is bound up with one of the most significant events of 1957, the launch of Sputnik 1 on 4 October 1957, and the tale of Farnborough’s involvement is told in detail by one of the key participants, Desmond King-Hele, in his book A Tapestry of Orbits. The short version is that with the aid of a large radio antenna hastily erected in a nearby field, and some calculations of Robin Merson, within two weeks they had an accurate orbit for Sputnik 1. This allowed them to estimate the density of the upper atmosphere and (after Sputnik 2) the shape of the earth. Robin Merson became an expert in practical numerical analysis and orbit determination.
Merson’s paper explains clearly the structure of the elementary differentials ... and, crucially, shows how they are in one-to-one correspondence with rooted trees. He also introduces various basic operations on rooted trees. This development, perhaps regarded initially as a bookkeeping device for finding and keeping track of the different terms, has over time become central to the combinatorial and algebraic study of B-series.
As it happens, the required mathematics and structures had already been discovered a century earlier by Arthur Cayley in 1857.
... Cayley needed trees for exactly the purpose we are using them here—to keep track of how vector fields interact when applied repeatedly to one another—and this purpose was then forgotten for a hundred years. The need for better numerical integration methods arose quite soon, towards the end of the 19th century, and the required tools for a complete theory were already present, but they had been forgotten."
The paper goes on to explain the connections to pre-Lie algebras and work by Vinberg, Gerstenhaber, and several other contemporary researchers. However, it doesn't mention the work of Charles Graves in 1857 on iterated operators of the form $g(x) \frac {d}{dx}$ (see page 13 in The Theory of Linear Operators ... (Principia Press, 1936) by Harold T. Davis).
A: Heaviside's operational calculus, used by electrical engineers to work with differential equations, predates its mathematically accepted justification by decades. The same can be said about Dirac's delta function, which is used together with it. Of course, to some extent the operational calculus is a repackaging of the Laplace transform, but that is not all there is to it.
One might argue that in this case mathematicians' splendid isolation worked the in the opposite direction.
A: Berry's Phase:
Berry's geometric phase in quantum phenomena, described in his survey article "The Quantum phase five years after", one instantiation of which is the striking Aharonov–Bohm effect, was instantly recognized by Berry Simon as facilely characterized by Hermitian line bundles and Chern classes as described in Simon's "Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase". For a prior technical application and verification of the initially controversial A-B effect, see Tonomura's review "The Aharonov-Bohm effect and its applications to electron phase microscopy".
A: I was once told that robotic engineers rediscovered and studied articulated systems, a feat already accomplished by Italian geometers 100 years
earlier if I am not mistaken.
Perhaps somebody can confirm this? (Wikipedia's article on 'articulated robots' is of not much use.)
A: I'm surprised no-one has mentioned general relativity and Lorentzian manifolds.
Einstein needed a general geometric theory of curved manifolds of arbitrary dimension in order to be able to model spacetimes in general relativity, only to find that Riemann had sorted all this out many years ago.  Riemann's work on Riemannian manifolds carries over to Lorentzian and pseudo-Riemannian manifolds with some minor mathematical modifications, although these modifications have important physical consequences (see here, for example).
A: My recollection is that the Finite Element Method was invented and used by engineers (civil engineers?) long before the functional analysts got involved and gave it a rigorous mathematical basis.
