Wow, that first hand was wild. Great time for quads. Bad time for anything else.
By my calculations, the odds
of getting the same hand at least 3 times out of 6 hands are roughly 2500:1 against, i.e. approximately 0.04% chance of happening.
The question remains, of course, are my calculations correct? I'll show my math in case anyone else wants to give this a go as well.
There are (52 * 51) / 2 = 1326 possible starting hands. Thus, the chance of getting one hand the same as another (i.e. two hands the same) is 1/1326. The chance of getting two hands the same as another (i.e three hands the same) is 1/1326 * 1/1326 = 1/1758276. The number of ways to order these three hands out of six hands dealt is 6! = 720. Thus, there are 720/1758276 possible ways to get at least three hands out of six the same as each other.
I hope I didn't botch that too badly!
In any event, it's definitely a rare event!