In my personal experience, I also had to sign a non-compete during on boarding which restricts me from working at certain companies. I have witnessed cases where my client wanted to hire an employee where we had a non-compete with. In many cases if the client and upper management come to agreeable terms and then it is possible. Although, these cases are seldom and require the employee to prove himself very valuable but they are possible.

There are many states that do not recognize the non-compete or the agreement is not enforceable in that state like California, Florida and etc. The reason for it is that it can be severe restrictions on competition and might make finding work impossible. For example, it is enforceable in New Jersey if the agreement pass the 3 tests specified by the NJ Courts.