When the dimension is an a priori fixed constant, then integer programming feasibility (the existence of an integer point in a polyhedron) can be decided in polynomial time. If the dimension is not fixed, then the problem is NP-complete.
Intuitively, what is the difference between the two problems?
One naive question would be: Why can't I compute the dimension first an then run the specialized algorithm? One answer here seems to be that in principle there could be one structurally different algorithm for each fixed dimension so that the joined algorithm (which must include all fixed-dimension algorithms) would be of infinite size.
Another answer I can image is that the exponents in the actual complexity depend on the dimension, something like $O(n^d)$ which would then be polynomial-time in fixed dimension. At least from what I read on Wikipedia about the LLL algorithm (going into this proof) this is not the case here.