I wouldn't say "always" (thought it does make a more dramatic effect), but often, yes. I often speed up when someone passes me. Rationally, when I see someone passing I look at my speed and if I see that I am going a bit slow I will speed up.

But irrationally:
1) socially conditioned sense of competition
I am not competitive, and whenever I find myself in that scenario consciously I refuse to play, still if I don't pay attention it sneaks up on me. I sensed a little competitive reaction from myself on the road more than one occasion.

2) imagined judgments
I found myself reacting to being passed as if I was being judged to be slow, or pokey, "another woman driver", etc. Amazing how much I believe I can read people's minds, eh?

It's all rather entertaining to see what kind of crap passes for thinking when I don't pay much attention.