Can part of a tampon break off while in use? I used one last night before bed and woke up with no period until later that night. This is not normal.

Yes. Just about anything can break, although tampons normally don't. With that said, I suspect you want to know what to do next, right? 2 choices: check inside yourself - you can't lose anything inside your vagina b/c it's like a pouch. Or go to ER or see your Family Doc/GYN/Planned Parenthood for someone else to take a look see. Of note, irregular menses are common but so are STDs/STIs w/o condom use.