Thinking of primary alcohols made me mistake primary alkylamine with 1-alkylamine, sorry about that.

Yes, you are correct, it is possible and it has to proceed initially via imine. NBS first N-brominates the amine, sodium methoxide eliminates the HBr from the N-bromoamine to give imine, which in turn undergoes an acid catalyzed hydrolysis to give ketone. I once attempted NBS bromination of a secondary amine followed by HBr elimination using DBU as a hindered base but I never got the reaction to work.
Patent number: WO 2004094371