NP complete problems that are solvable in polynomial time if the input (e.g. number of variables) is fixed?
In parametrized complexity we solve the problem by fixing some parameter (say $k$).  If we are able to solve some problem in $f(k) \cdot p(n)$ time, we say the problem is fixed parameter tractable in $k$.  Here $f(k)$ is just some computable function.  There are lots of NP-hard problems that are FPT, however, there are many problems in NP that are believed to not be fixed parameter tractable. 

If by fixing some parameter we can solve a problem in time $O(n^{f(k)})$, this problem is said to be in XP.  We believe that XP is not equal to FPT (just as we believe P $\neq$ NP).  But there are also lots of problems between these two (FPT and XP), and we have defined a hierarchy (actually several), one of those being the W-hierarchy.  In the W hierarchy you have reductions like reduction in NP-complete classes, except, we are not looking for polytime reductions, we just need an FPT reduction.  The class W[0] is the class FPT.

These are some samples in different classes of the W hierarchy:


Vertex cover is FPT (so is vertex disjoint paths on undirected graphs)
Independent set and Clique are both W[1]-complete
Dominating set is W[2]-Complete.


These are another level of complexities to classify NP problems in more precise way and if you want more you can look at this paper.

And if you want even more is good to read Grohe and Fomine's book

And finally:


  Is this always the case for problems that admit an
  FPTAS/pseudo-polynomial time algorithm such as Knapsack


Not necessarily, it's known that if the problem has FPTAS then it's also FPT (which is obvious), but there is some works on the relation of PTAS and XP, but there isn't very tight relation between PTAS and W hierarchy (at least I don't know at this moment).

Also in some cases may be we fix some different parameters, e.g: length of a longest path in the graph is bounded and size of a solution is bounded (e.g in feedback vertex set), ...