Last time I switched jobs I gave notice after I had cleared the background check and drug screen and I still ended up having an unplanned week off because the new company didn't get their act together in time! That wasn't an issue for me - I wanted some time off - but they had been pressuring me to start right away so I had planned to jump right from one job to the other. So when they then called and said "oops - sorry. We aren't ready for you yet!" it was a bit of a shock. The delay was only a week, and I really enjoyed that week off, but I can see where it could end up stretching out even longer if something unexpected happened. So it's probably best to wait to give notice until you have a firmly established start date at the new place.
Back during the .com bubble burst, I knew folks who had an offer letter in hand, had accepted and were due to start in a few days, but then the company called up and basically said "we take it back - the offer is rescinded"! That's less likely to happen these days, but you still never know...That taught me to cover my you-know-what as best I can and not show my hand to my current employer any earlier than necessary. (My personal horror story was being laid off along with my entire division - from the VP I reported directly to all the way down - just a month after I started. Who the heck is hiring people one month, then laying off half the company the next month?! Ah, good times....NOT)