Since I know quite some people who exactly work on this kind of problem of finding special classes that can be solved in polynomial time, my impression is of course different.
But I think it can be said that if some NP-hard problem is very important in practice and there is no easy way to to get around this problem (i.e. it will also be practically relevant in, say, 15 years), the problem will for sure get a lot more attention.
This is also the reason why some NP-hard problems are much more researched than others.
Sometimes it's also helpful to look into approximation algorithms, e.g., a good LLL implementation can work wonders for certain problems. But heaven forbid someone obtains an inapproximability result for your problem, then you're really in trouble.
This is not (necessarily) true:
For example, there exists a great approximation algorithm (Goemans-Williamson algorithm) for MAXCUT in graphs with non-negative edge weights.
On the other hand, when negative weights do occur, one can show (unless P=NP) that there exists no polynomial-time approximation algorithm for which the approximation guarantee is a constant factor times the optimal solution.
But since the Goemans-Williamson algorithm is a great algorithm (if the Unique Games Conjecture holds, and P != NP, it has the best approximation guarantee that any approximation algorithm for MAXCUT with non-negative weights can get in polynomial time) nobody "forbids" you to use it in situations where also negative edge weights can occur. You will loose the approximation goodness guarentee, but in practice, this algorithm nevertheless gives good results in this situation, just not certified good results.
Overall, it's annoying to tell whether an NP-hard problem is always really hard, or if ~all practical instances can be solved with a clever heuristic. It doesn't help that most problems receive little attention (e.g., to find solvable special cases) after being labeled NP-hard.