In the play, "Romeo and Juliet", is it possible that a love story of this magnitude could take place so quickly?
Romeo & Juliet is first and foremost a play. Moreover, it was written well before a fashion for realism in fiction of any kind. Given these parameters, we should expect a fictional/dramatic world in which the improbable simply happens, an experience closer, perhaps, to grand opera rather than drama as we have come to know it today.
There is - as with most of Shakespeare's work - an emotional, human truth that transcends questions of realistic possibility. The love, the jealousy, the warring families are 'real' enough as human experiences, as of course was the common practice of very young daughters of aristocratic families being promised in marriage for political purposes, and denied the freedom to fall in love and choose for themselves. The play takes these themes and produces a tragedy (not a romance!) which is partly the result of youthful rashness and mis-timing. Besides, the very young notoriously do fall in love and make declarations they mean with all their hearts at the time...
You might consider modern dramas, especially film, in which the scenario is pure fantasy, but which nevertheless evoke powerful, believable emotion. It doesn't matter that Nemo is a cartoon fish, or that Superman is a comic book creation: Nemo's little guy heroism, and Superman's grief at losing Lois, appeal to us on another level than the one that demands social realism. Watch R&J again!
Shakespeare didn't focus as much on timelines (you'll find many inconsistencies) but rather more on events and the feelings involved. Sometimes instant love is really instant lust in disguise but having said that, some couples fall in love deeply and quickly and it lasts. We have to remember that R & J are very young and that it may just be an intense infatuation, which happens at that age, although that is not to discount their feelings.