That is particularly the case where an employee has been fired (assuming that, as in most cases, the employee wasn't fired "for cause").
But there is an important exception to this rule: the employee choice doctrine.
"What's that?" you ask.
The employee choice doctrine states that even when an employee is fired, the non-compete or non-solicitation clause in his employment agreement will remain enforceable if the employee agrees to receive post-employment severance or other benefits in exchange for his honoring the non-compete.
And under those circumstances, New York's courts have held that the non-compete doesn't constitute "an unreasonable restraint upon an employee's liberty to earn a living" because the employee is then making "an informed choice between forfeiting his benefit or retaining the benefit by avoiding competitive employment (Kristt, 4 A.D.2d at 199, 164 N.Y.S.2d 239)."
There is a critical point to be made here. Once the employee signs off on this new agreement, the non-compete will be enforced - even if the agreement is terribly unreasonable.