Keep in mind state rules and terms of accepting severance, its a state by state legal situation but in some it means you waive your right to the unemployment claim, others it means you can't file unemployment until a specific date listed in the severance agreement, in others you get both severance and unemployment, just a friendly reminder to read the terms of the severance agreement.
For Example in Texas When you file for unemployment benefits, your last employer receives notification and an opportunity to oppose your benefits. If you receive a severance package, the content and terminology, along with employer opposition, determines whether or not you receive unemployment compensation benefits.