I agree with timtracker, but would also add that the way you record the quarter in which a birth, marriage or death occurred is entirely up to you really, as long as you know what your convention means. I did go through a phase of recording them as you've suggested, e.g. "bef 31 Mar 1890", but settled on "abt Mar 1890" as it's easy and I know what it means.
Some inaccuracies are bound to creep in when using these records, simply because of the range of dates which they may cover, but this range is at least finite. The worst case of inaccuracy could actually be illustrated by the example you quote, as a person registered in Jan to March could have been born as far back as November in the previous year, but again, I accept that this might happen and allow for it when finding further information for the person concerned.