Changing insurance while in treatment?

Currently a friend of mine is running into this situation:

Her current insurance ends by 7/31 and her next insurance starts at 8/1. However, if she is in a long-term medical treatment from now to September, for example, how does the insurance work in this situation? Will the first insurance pay all incurred bills till 7/31 and the second insurance pay the bill incurred after 8/1?

Thanks for any help!