"In some cases you apply for unemployment immediately and severance comes after a certain period or time."
Actually I don't believe that is correct. CA severance is paid once you sign all paper work. It literally can be paid out in full in the next pay cycle (at least what I heard from others in the past) This next part likely varies by state, but often unemployment can only start once severance "runs out". So if your paid 4 weeks severance starting May 15 while CA will pay it in full.. However its basically like them keeping you on the payroll until June 15. So then you probably cant file UE until June 15