Non-competes for an unspecified amount of time are generally not enforceable in NY. You should retain experienced employment counsel to help you negotiate a reasonable non- compete.
The information provided above is for general purposes only. It does not constitute legal advice or create an attorney-client relationship. Seek competent legal representation, because the facts of each case are different.
Non-competes are very fact specific and there are too many missing facts to get an accurate answer. While I don't like to give vague answers, without spending 20-30 minutes at least discussing this and seeing the actual agreement, I would be guessing. I have handled numerous non-compete cases and each one is different and turns on the specific facts at issue.
The validity of a Non-Compete centers around the restrictiveness in terms of both time period and geographic area covered. The clause can either be thrown out or scaled back if challenged and found to be unreasonable. I'd suggest you discuss more details on your specific situation with an attorney before going any further. Being proactive in resolving this can help you resolve the fear of enforcement and also ease your mind as you take on students. Best of Luck.