take the kitten to your own vet soon after adoption! My Clyde was born in the shelter where I volunteer, so I knew more about his history than most adopters are ever able to find out. He seemed fine except for mild lingering "kennel cough"... but in fact, he was very sick indeed (pneumonia and pleural effusion), and if I had not taken him to my own vet early on, he probably wouldn't have made it.
The irony is that I later found in his shelter records where a visiting vet had noted that he had not been able to hear Clyde's heartbeat... a clue, you would think, but he just let it go. As it turned out, the reason he couldn't hear Clyde's heart was that it was being pushed out of place by a huge fluid buildup in the wall of his nearly-collapsed right lung!
It's amazing how sick a kitten can be and still seem okay. Please do let your own personal vet see your new baby, just to be safe!