What constant ensures hyperbolicity of Dehn surgery? I am interested in showing that certain knots having a surgery description are hyperbolic. Unfortunately I have not had time yet to read Thurston's work, so my understanding of this is vague. But from reading some papers, it seems that the following is correct (so the first question - is it correct?). Given a hyperbolic link, there is a constant, such that filling a subset of components with slopes with denominators bigger than this given constant, one always gets a hyperbolic manifold (complete, finite volume?).
So my real question is: can we explicitly find this constant? We may or may not be given the volume of the initial link.
 A: This is really just a long comment expanding on Bruno's answer. However, I wanted to include some handy references. 
The approach that will probably be most applicable for the purposes of this question are derived from Agol and Lackenby's 6 theorem.  In Exceptional Dehn surgery on the minimally twisted five-chain link, Martelli, Petronio, and Roukema check the hyperbolicity of infinite families of (n-1)-cusped manifolds coming from surgery on an (n)-cusped manifold is handled quite nicely. Also, Ichihara and Masai's Exceptional surgeries on alternating knots uses similar methods, but applies them to a different set of manifolds.
EDIT: In the paragraph below, it is assumed that the framing of the peripheral torus and so the labelling of the surgery slopes is fixed. In terms of the question being posted, it would make sense to use a geometric basis, i.e. take a pair of the shortest two linearly independent elements. As Ian Agol's answer points out, the blackboard framing of one cusp of a link and the geometric framing of the same cusp can be wildly different. 
Let me expand upon Martelli, Petronio, and Roukema's argument because I believe it will be most directly applicable for you. Suppose $M$ is a three cusped manifold and $M(\alpha_i, \beta,-)$  (i.e. filling the first cusp along the slopes $\alpha_i$  and the second cusp along $\beta$, and the third cusp is unfilled) is non-hyperbolic for an infinite number of distinct $\alpha_i$. Then $M(-,\beta,-)$ is non-hyperbolic. Similarly, there are only finitely many $\beta$ to check for non-hyperbolicity if $M$ hyperbolic. The code (with documentation) associated to the paper is available on Bruno's website and fairly easy to implement and adjust (after installing Snappy). 
Finally, for there were tricks that one could use other than the 6 Theorem. 
One method is involves keeping track of the distance between pairs of non-hyperbolic fillings. Cameron Gordon's surveys "Dehn Surgery on 3-manifolds" (available the book Low Dimensional Topology) and Dehn filling: a survey (the later deals mainly with closed manifolds) are good places to dive in the literature if you are trying to understand this approach. 
A second method directly applied to knot complements is Kadokami's Hyperbolicity and identification of Berge knots of types VII and VIII. This method involves showing a set of knot complements known to be atoroidal are not torus knots showing that the Alexander polynomials of the knots in questions do not overlap with torus knots. Then Thurston's work establishes the hyperbolicity of the knot complements being considered.
A: Just a remark: there is no universal such constant, even for links with bounded volume complement. 
For example, there are infinitely many links with the same complement as the Whitehead link, e.g. 

Then $\frac14$ Dehn filling on the blue component gives the trivial knot,
which is not hyperbolic (at least not with finite volume). Adding twists
to the red component, we may obtain examples with arbitrarily large 
denominator non-hyperbolic Dehn fillings. 
A: Given a hyperbolic link $L$, Thurston's Dehn filling theorem says that there is a finite set of "bad slopes" on each component of $L$ such that every filling avoiding them is hyperbolic. A slope is just a rational number, and since the bad slopes are finite in number, of course if all denominators are sufficiently big to avoid them the result is hyperbolic. So your guess is correct.
A simple way to find these "bad" slopes is to download SnapPy, practice a bit with this user-friendly nice program, then draw your links. If you have some specific Dehn surgery parameters to test, just give them to SnapPy. If you look at these finite sets of slopes to avoid, you must look at the cusp shapes of the boundary tori. The cusp shape of a torus is a complex number $p+qi$. You can then select a cusp section of some area $A$ manually, and from the three numbers $p,q,A$ you can very easily list the finitely many slopes having length at most $6$. These are the "bad slopes" to avoid on that component, by Agol-Lackenby's 6 theorem.
If your link is a knot, then it is conjectured (and almost proved) that if the denominator is >2 then the result is hyperbolic. For links things are however more complicate.
