The only exception, which drives my use of the words "In general", that I know of involves Photoshop Elements which can be used on multiple platforms. All other licenses are only good for one OS type.
You are better off contacting someone at Adobe directly if you want to see if there are any exceptions pertinent to your situation, though yours is a fairly common one, so chances are the answer will be that you need to buy a new license.
Which Adobe product exactly?
Photoshop Elements, Premiere Elements and Lightroom are all multi-platform serial numbers - work on both Mac and Windows.
Serial numbers for all other products are operating system specific.
(Note: Adobe Creative Cloud products are all multi-platform too but they don't require a serial number. Adobe ID only.)