Having an instant connection with people doesn't happen all that often, considering how many people I'm 'friends' with (here, at my previous job/town, etc).
I can tell whether or not someone will be a genuine friend within a very short amount of time. But even then, some of those more genuine friends may not last indefinitely, and as time goes on I may notice things that don't work well between us, which will ultimately stop us from ever reaching a true 'best friend' level. I've only had four of those in my life, all of which are in the past. I have a small handful of very close friends, and a few more casual friends, while the rest are mainly just people I will share polite small-talk from time to time or a good laugh about something we have in common.