Fanella, in the contrary a rescue Havanese's pareantage will most probably be known and documented, if that's what you want.
There are two options available to you - non breed-specific rescue or a rescue dog from a breeder (as far as I know there is no stand-alone Havanese rescue organisation).
A rescue which takes in all breeds is unlikely to see a Havanese - it's possible but rare. Should they do so they may not have a history as the dog may be a stray but it would be far more likely that he would have been handed in by an owner. Whether he had a tracable parentage would depend on where that owner had obtained him. If he'd been purchased from a reputable breeder then he will be fully tracable, if from a backyard breeder (eg the man down the road who has bred his bitch with a pal's dog/dog found on the free ads just to make a fast buck without regard to breed lines or health checks) then there will of course be no parentage worth checking. On the positive side to this, IIRC Havanses are not known for a mass of potential health problems - eyes being the only real concern - so this need not necessarily be a barrier to adoption. Personality is down to upbringing and treatment more than breed or parentage too, IME.
HOWEVER... most Havs end up back with the original breeder if the owner can't keep them so you will know everything there is to know about your adopted dog and his parentage. Decent breeders ALWAYS- no ifs, no buts - INSIST that a dog is returned to them if the owner cannot keep him, just as a decent rescue does. (I must say here that IME there are more decent rescues out there in this and all other respects than there are decent dog breeders).
I have just taken the liberty of emailing a couple of breeders and have already got a reply from one of them. That lady has no rescue dogs at the moment but is happy to contact other reputable breeders whom she knows personally and ask around on your behalf. However she will need this from you first hand (KC regulations) so please can you pm me and I'll put you in touch. She is one of those who insists that any dog which the owner cannot keep must be returned to her.
Hope this helps, I'll pass on all the info and my email address in case you need more details, just drop me a pm. :)