I suppose you could do a bard as a half caster, though I see no compelling reason to do so. Also, there is no "Divine" or "Arcane" distinction in 5e, just spell casters and how they get their spells. In addition, I would point out that the two half casters we have posses strong class abilities and features outside of casting to fall back on, and are fighters in all but name. The Paladin has smites that his spell casting fuels, while the Ranger has the "we don't want to anger the anti-4e crowd with martial or mundane abilities that are too magic like, so anything that might hint at a supernatural ability we will put in the spell casting system" going for it.
The Bard, while having a hodge-podge of abilities outside spell casting in earlier editions, does not have the strong, fighter-like chassis to build on (unless you're talking about he 1e bard, which started out as a fighter, then druid, then magic user. Or was it the other way around?). Also keep in mind that, despite this hodge-podge of non-spell casting abilities used over the years, the Bard has never been, shall we say, universally acclaimed. The whole jack-of-all-trades-master-of-none concept that the Bard has sometimes taken has not traditionally done that well in D&D, though it has it's fans. None of this is insurmountable, but why bother?