At the end of the The Crucible, John proctor tears up the confession that would have saved his life. Do you find this act believeable?

Absolutely, I do.  We see John's ambivalence about lying to save his life in his conversation with Elizabeth.  He says, "I have been thinking I would confess to them, Elizabeth [...].  What say you?  If I give them that?"  However, he cannot look at her when he asks it, and when people cannot make eye contact, it often signals that person's shame.  Further, he asks her opinion because he respects her as a good, honest woman.  If she can say that she would give a lie to save her life, it would make John feel less guilty about doing so himself.  He explains further,

I cannot mount the gibbet like a saint.  It is a fraud.  I am not that man [....].  My honesty is broke, Elizabeth; I am no good man.  Nothing's spoiled by giving them this lie that were not rotten long before.

Ever since he committed adultery, John has felt like a corrupted person, one who can never regain his goodness.  It has tortured him.  Now he says that since he is already ruined, nothing more is damaged by another sin.  If he goes to hang like one of those people who are truly innocent of sin, it would be another kind of lie.  He is trying so hard to justify this to himself.

Elizabeth tries to tell him that she cannot judge him either way, that "it come to naught that [she] should forgive [him], if [he'll] not forgive [him]self."  She assures him that whatever he does, "it is a good man does it."  She believes him to be a good man regardless of which path he chooses now.  Finally, most importantly, she tells him,

Do what you will.  But let none be your judge.  There be no higher judge under Heaven than Proctor is!  Forgive me, forgive me, John -- I never knew such goodness in the world!

Initially, this makes him want to keep his life, to stay alive and be with her and their children.  But then her words, I think, have time to sink in while he makes his confession, answers the judges' questions, considers whether or not to sign the confession, and explain why he does not want to.  When he tears up the document, in agony, it brings to fruition many, many months of self-doubt and recrimination and shame.  John finally sees a way to reclaim his goodness, his sense of personal integrity, and he says, 

You have made your magic now, for now I do think I see some shred of goodness in John Proctor.

He ultimately realizes that he doesn't have to be a saint to be good.  It isn't about being sinless but choosing as well as he can in this moment, at this time.  I find it extremely compelling, and, because we have seen Proctor's emotional and spiritual journey up to this point, it is quite believable.