The answer, as it often is in the legal world, is it depends.
All employees have a duty of confidentiality, and higher level employees may have fiduciary duties which will further limit their ability to compete. Confidentiality will prevent any employee from using confidential information for the benefit of themselves or third parties. For example, you cannot use your knowledge of your former employer to deliberately undercut them, or your knowledge of their contracts to strategically target their former customers.
Additional post-employment restrictions can be imposed by employers, usually in the form of non-competition covenants or non-solicitation covenants. However, just because it’s in writing, doesn’t make it enforceable – even if you signed it. Courts can strike down any restrictive covenant it deems unreasonable.