No no no no no.... No.
The Non Compete agreement means that you won't share secrets with their competitors or any competitor to enter the market from the time of your signing.
I can have a job now at another hospital or college. If they release something that is traced back to me sharing trade secrets then I can be sued.
Like Chris said, you sign them ALL the time in I.T. The difference between the IT one and perhaps yours (or another) is that in IT we have knowledge of trade secrets and more (Emails, for instance) that can easily be used to bring a company out of business.
In normal ones, they just don't want you to either work for their competitor during their product cycle (Anywhere from 30 days to many many years) or they don't want you to work in their area of business for a specific amount of time (again, their business cycle) You can't start your own business in their line of work. So, you can't learn all sorts of shit, get intimate with their clientele and then run off and do it independently. Their time that they forbid this is decided not by them but by a regulating body. In my business (the one which I own a permit) my business cycle is 30 days. The moment I filed for a patent under my old one, it became 2 years. (R&D, sales, and life cycle of product). They chose those times because I held a patent in an automotive design.
Now, I change oil and try to stay on the down-low every year with my money. At least for now. When I get back on my feet, I'll have a few things laying around the workbench that I don't want others to see. So I'll also have a non-disclosure agreement with any employees or constituents.
This stuff won't prevent you from getting a job anywhere. Trust me on this. It only gives them the right to sue you if your new employer copies one of their ideas, and they can prove that it was you that gave it to them.