It really really depends on the company,

When my hu*****and was laid off, the HR of his company (a big Internet company) told them orally (because their written letters didn’t state clearly whether it’s allowed or not): if you find a new job before the official termination date, we will be happy for you but just don’t tell us and we will not actively chase you and find it out. If you find a job after you get the severance package, that’s great too, so go ahead for the new job and while enjoying the severance package. So some of my hu*****and’s ex-coworkers took both severance package and a new job after the official termination date.

So do check with the company and hear what they say. Some companies are very tolerant with this but some are not.