3-COL with a restriction on a number of vertices colored with a particular color
For a fixed parameter like 50, the problem can be solved in "polynomial time": guess which vertices to color A, then solving a 2-coloring problem for the remaining vertices.  The running time will be something like $O(|V|^{52})$, since there are $O(|V|^{50})$ ways to choose a set of at most 50 vertices; and 2-coloring can be solved in $O(|V|+|E|)=O(|V|^2)$ time.

Of course, this solution is solely of theoretical interest.  An algorithm whose running time is $O(|V|^{52})$ is technically polynomial-time, but it's completely and totally infeasible in practice for any interesting graph.

In contrast, if the parameter 50 is part of the input, then the problem remains NP-hard (consider what happens when you pass in $|V|$ as the value of this input; then you have the standard 3-COL problem).  This highlights the difference between a value that is fixed vs is an input to the problem.

Perhaps a simpler example to illustrate that difference is vertex cover: vertex cover is NP-complete, but if I ask you whether there exists a vertex cover of size $\le 50$, that can be done in polynomial time, for similar reasons -- the running time will be $O(|V|^{50})$ or so.  In particular, if you want to know whether there exists a vertex cover of size $\le k$, that can be done in $O(|V|^k)$ time.  When $k$ is a fixed constant, this is polynomial time; but when $k$ is part of the input, then this is exponential time (it is $\ge 2^k$).