Not usually. The post-termination exercise period begins at the end of employment. Most stock plan documents state that your vested options remain exercisable for a specified period after the cessation of employment (or, alternatively, you must exercise them before termination). It is not affected by the severance-payment period unless your severance agreement or stock plan specifically states otherwise.
Example: Your official termination date was September 19, but you received a severance package through December 31. The plan document allows you to exercise your vested stock options for 90 days after termination (i.e., until December 19). Don't confuse the length of your severance package with your post-termination exercise window.
In some situations companies may continue to pay your full salary, health benefits, and pension benefits during severance, or may use you as a consultant. Confirm with your company whether this would be considered continued "employment" under the stock plan. If so, the post-termination rules are not triggered until the severance-payment period (or consulting period) ends.