Playing a Nash equilibrium strategy is not, contrary to how it sometimes presented, in general the best strategy. Recall that the definition of a Nash equilibrium is that, each player, if they were to know the other player was playing the Nash equilibrium, would continue to play in the Nash equilibrium.
That doesn't say it's the best strategy at all! It's an equilibrium, not an optimal strategy.
There are notions, such as of course the notion of dominant strategy, that are objectively the right thing to do. But a Nash equilibrium is a bit different. If both players play in a Nash equilibrium, then they are both fine with the other player finding out their strategy. That is to say, they have a certain sort of assurance: they might as well proclaim their strategy publicly, even -- and they don't have to worry about the other player somehow predicting what they are going to do and thwarting it. It's an "equilibrium" because if the state of the game evolves, by both players learning something about the other person's strategy, they still both continue to play the same way. In iterated games, this means their strategies do not change from round to round.
Nash equilibria arise naturally, therefore, only in situations where information is guaranteed to eventually leak out. The classic case is in studying evolution, where (we might assume) species A adapts and evolves to know the behavior of species B just as species B evolves to know the behavior of species A, and in the end the state of information converges to both species knowing the other's strategy. Hence the equilibrium.
On the other hand, there are numerous examples where players would not want to play the Nash equilibrium:
In games of asymmetric information. For example, you are aware of some benefit to you in playing $X$ that your opponent isn't aware of. But, just because that is your best strategy given what you know doesn't mean your opponent knows it's best for you, and so there's no reason the Nash equilibrium condition would hold for this move.
If you think you can predict your opponent's strategy. If you can predict your opponent's strategy (with even above-random accuracy) and you think she will not be playing the Nash equilibrium, obviously then you are in no rational obligation to play it either.
I should say I think there are arguments for Nash equilibria being a good strategy to play just by default. Perhaps, the argument goes, since you don't know what your opponent's going to do, and she doesn't know what you're going to do, you "might as well" just play the Nash equilibrium. The argument isn't saying there's any good reason you have to play the Nash equilibrium; of course, it's not guaranteed to be your best bet. On the other hand, since you know nothing of what your opponent will play, no other move is guaranteed to be your best bet either (assuming you have no dominant strategy). So, in the absence of any real surety with respect to what you should do, it seems reasonably "safe" to do the Nash equilibrium move. And at least if both players go with this "safe" option, then you can rest assured you couldn't have played any better than you did, given how your opponent played.