The only cases that are clear cut is if the "first marriage" is with a convert/non-baptized person. In that case, no sacramental marriage existed, so a catholic marriage can happen. All other cases are pretty hairy.
How does being a convert enter into it?
Cases where a Catholic did not follow canonical form (and was not dispensed from it), commonly referred to as "marrying outside the Church", are pretty clear-cut. The only thing that is missing, is the Church having all the documentation that things happened that way, saying "okay, it's proven, Romeo married Juliet outside of canonical form, they weren't dispensed, nobody ever got a sanatio in radice
, therefore there was never a marriage". Even if Romeo and Juliet were both Catholic, and sought out a justice of the peace, there would still be no valid marriage. If a non-diocesan traditionalist bishop assembled the facts in front of him, and had all relevant facts proven, then it's hard to see a problem with that.
The more subjective cases (too immature, had a psychological problem, couldn't balance a checkbook, didn't get along with the neighbors, what have you) wouldn't be an issue, because prior to V2, those weren't grounds for annulment.