It is not my field, but I would like to mention this example anyway since when I learned it some time ago I was very impressed. In quantum field theory, in particular in quantum electrodynamics, one assumes existence of the whole theory, namely operator valued functions on $\mathbb{R}^{3+1}$ which should satisfy various properties, e.g. equivariance under the Poincare group, equal time commutation relations, existence of in and out states. However existence of such objects is not proven in physically interesting situations, e.g. for quantum electrodynamics in 4d. For me, as a mathematician, it was quite shocking and took a long time to realize that such advanced and non-trivial objects are only believed to exist, and were not constructed even in any non-rigorous sense. Moreover as far as I heard, now it is believed that some of these theories even should not exist (!), but they worked well so far since they are expected to be good approximations to more sophisticated (probably) existing theories.