Of course it is possible, but it does have a rather low success
rate. Even if both parties were to be mutually accepting of the
differences, the parents of each person might disagree. If
conversion by any one party is required, it is best to see if the
reaction of the parents show even the slightest disapproval- it is
a strong indicator that they most likely will not give their
blessings should there be a marriage.
When such a time comes, both parties have to be rationally aware
of their situation and come to terms with it. It is better to nip
something in the bud before making it escalate by being blissfully
unaware of the consequences.
To put things into perspective, let's say Muslim girl (MG) is
going out with Christian guy (CG). They've been dating for say, 2
years and have come to the 'meet the parents' stage. Assuming CG is
willing to give up his religion for the girl, both parties have to
see whether CG's parents are receptive to the notion or idea. It
will be especially difficult if they are staunch Christians and
brought up CG to be a staunch Christian as well. If they show
horror, disapproval, or even pass hints that they would not like it
if their son is to turn Muslim (e.g. passing comments like he
should remember to go to church even after marriage), then it is a
very good indicator that they are not going to be receptive of the
conversion.
This is then where both parties have to discuss the situation
and decide if they want to try to push their luck or to start
cutting the relationship off. It's better to cut it off earlier
than later, because things might get complicated later on.
Therefore even though it is hard, it is the most pragmatic solution
available if both parties want to save themselves from more
heartache and time/money wasting in the future.