The same cutscenes that introduce us to Hilda also imply that these 'indiscretions' are regrettably common. Heck, even Count Fortemps has a bastard son, and he's presented in the game as an exceedingly noble man. It seems to be just an accepted part of being an Ishgardian nobleman, the assumption that they're gonna dally with the attractive serving wenches from time to time.
Given that all nobles seem to be Elezen, and that the Hyur population of Ishgard are pretty much all commoners, it's odd that half breeds aren't more common. You'd think that the whole taboo of interracial couplings would make Hyur women all the more exotic and tempting for horny young Ishgardian noblemen. (The reverse gender situation would likely also be true - but I'd imagine that commoner women have a lot less power to say 'no' to avoid getting knocked up than the noblewomen would, so scandalous pregnancies would be less common among noblewomen than commoner women.) So, it's kind of puzzling that there aren't more racial combos around.
One explanation might be that these couplings are a lot more common than people like to admit, but the fertility rate is low when dealing with mixed races. Or, possibly, it's far more common for a mixed-race child to favor one parent's race over the other so strongly that they can easily pass as a pureblood, with obvious half-breeds like Hilda being a rare exception.
The introduction of Hilda was a bit aggravating, to be honest, and opens up a whole big ol' can of worms that could have been avoided, much like the case with Lilisette in FFXI, another one-of-a-kind half-breed (Hume/Elvaan - yes, another "half elf"). They've opened the door to the existence of half-breeds, and I hope that they don't go the way FFXI did and never touch on the topic again... What's worse, before Lilisette was introduced, FFXI actually had snippets of dialogue that implied that mixed-race breeding DID happen, but that the child was always the same race as the mother, which would have been a tidy solution to the issue. Lilisette threw that notion to the curb - and never provided an explanation why.