First, the play takes longer than three days. You have to remember that Othello is sent to Cyprus and that journey alone, along with its preparations, probably took at least three days. However, at its core, your question is a good one. How can such a noble man fall into such a dastardly trap so quickly? I think one has to look at the character of Othello himself in order to find an answer. Although he seems intelligent and noble, Othello is a Moor and thus an outsider in Venice. As a military man, he does not possess great wealth and he is of a different ethnicity than most of the city. He probably has some feelings of inferiority because he lacks some of the traditional status symbols. He has just married the beautiful Desdemona, a woman of high social status, and, although this is a happy occasion for him, it also adds to his sense of inferiority. In addition, Othello suffers from epilepsy, a disease which was very misunderstood in Shakespeare's time. A seizure could take away Othello's command of his own actions and behavior at any time. This also had to add to Othello's lack of self-esteem, especially considering some thought epilepsy was a sign that one was possessed of the devil. Given his dark skin, epilepsy, and his father-in-law's disapproval of his marriage, Othello must be especially vigilant about his reputation. Any threat to that reputation could mean his ruin of both his marriage and career. This leaves him very vulnerable to Iago's plot.
It's funny that this question focusses on the time scale of the play. It's a key problem. Because the play, as you suggest, does happen in three days, but as ms-mcgregor suggests, it also doesn't. There's a double time scheme.
On the first day that the Venetians land in Cyprus, there are public celebrations, the drinking scene happens (with Cassio stripped of his lieutenantship) and Othello is with Desdemona all night (no chance to cheat!). The next day, Desdemona begs on Cassio's behalf, and Iago convinces Othello that she has cuckolded him. Lodovico arrives that afternoon, the "willow scene" occurs that evening, and Desdemona is dead that night. There is - within this logic - no time for Desdemona to have been unfaithful.
Yet Bianca says to Cassio that she's not seen him for a week. And it seems there is no real logic - bearing in mind Desdemona and Othello only got married the night that everyone went to war and left Venice - to Iago's claims of an affair.
Psychologically, I think the play is totally plausible: Iago praying on Othello's insecurities and prodding that jealousy until it expands, defeating all logic and doubt in his brain. Time-wise, it both is and it isn't.
Why the double time scheme? Who knows. Could be a mistake. Could allow Shakespeare to have his cake and eat it - a tight, quick tragedy, but also with enough time for the jealousy to develop. There isn't an easy answer to this one! But I hope this helps!