In Amsterdam, it happens all the time. In fact, I'm sometimes surprised when I'm out and don't run into someone I know.
I suppose it's a sign of living in a small city. People walk or bike everywhere, which makes it much more likely to come across actual human beings. And while there are a lot of cool places, people with similar tastes (like my friends and I) end up wandering around the same areas at the same times.
The upside should be obvious. You get to run into people you like, and it's a pleasant surprise. Plus it turns every trip out into a potential source of gossip; which is always good. And if you're like me, you answer the "we should do something sometime" with "okay, what are you doing on Thursday", and end up with something fun to do.
I suspect this is not unique to Amsterdam. I was visiting a friend in DC, and we ran into someone I know while going out to a movie. Odd because I don't live in DC, but perhaps merely yet another sign of how cities are way better than the countryside or suburbs.
A quick note about the word "acquaintances". To be more accurate I guess one should probably say "friendly acquaintances". The Dutch have a word which I think actually fits better, "kennisen", which basically means "acquaintances" but the connotation is slightly different.