Proton attached to N or O in HNMR spectroscopy
It depends on the condition of your experiment if a coupling of a proton of an amino or hydroxyl group with a proton attached to the $\alpha{}$-carbon is observable, or not.  As pointed out by the indicator set by @Mithoron, intermolecular exchange processes need to be taken into account.


If your alcohol (or amine) to be analyzed is very pure, and the deuterated solvent you deploy is both very pure, and dry, and free of acids or bases catalysing proton exchange, than you may observe $^3J$-couplings you expect to see.  For example a freshly opened sample of deuterated DMSO (used as solvent) may contribute to this observation.  See the peak data of the 1H NMR spectrum of ethyl alcohol in DMSO here.
If you aim to analyze an alcohol, not well dried, analyzed in deuterated chloroform or methanol that was frequently exposed to ambient (humid) air, then indeed instead of a triplet of an hydroxyl group  ($\ce{-OH}$)  attached to a methylene group ($\ce{-CH2-{}}$) all what may be spot is a broad singlet signal.  The same reference, 90 MHz 1H NMR in deutero chloroform provides one example.  (Note the chemical displacement of the hydroxyl protons altered, too.)