My girlfriend and I were having sex and little did I know, the condom broke. But not all of the sperm escaped. She also told me recently that she got her period roughly 5-7 hours later, while she was sleeping. Is there a chance she might be pregnant?

Yes, it is common if there is sex within a few days of the period, for the girl to still have the period. You will have to wait till next month and see if she misses it.

While it is not as likely, considering the millions of birth each year, some will happen just like that