This request has been made of me a couple of times as a buyer because the property my seller(s) were in simultaneous escrow to purchase was not going to close exactly on time. Both times it was for less than a month and I made sure it was more than worth my while to do this and that I was paid their future rent in escrow, AFTER getting the lowdown on their holdup from the escrow officer handling the concurrent transaction.
In both instances, their transactions closed on time or a little early and my sellers moved out in a timely manner, leaving everything fine – no problems at all. If handled by competent agents representing ethical seller-occupants, I think this arrangement could work out. However, I have always written offers to deliver the property vacant and would not agree to this or advise a buyer to agree to this if the occupants were tenants or the problems holding up the concurrent escrow involved appraisal or financing issues.