Angular momentum is a vector, not a scalar. In the case of hydrogen, this makes it possible for the total spin to be either 1/2+1/2 or 1/2-1/2.
By the way, in most cases we don't care at all about the total spin of an atom, because the hyperfine coupling is very weak. We care about the total spin of the nucleus, and the total spin of the electrons.
It is in general beyond the state of the art to predict with perfect reliability, from first principles, the ground-state spins of all nuclei. I'm a nuclear physicist, not an atomic physicist, but I believe the prediction of the total electronic spins is much more tractable, and can basically be done using Hund's rules.
In nuclear physics, for an even-even nucleus, the total spin is always zero. For an odd nucleus, if you know the shell that the odd nucleon is in, then the rule is that all the angular momenta couple so as to cancel, except for that of the odd nucleon. I.e., angular momentum is minimized, which is the opposite of Hund's rules. In an odd-odd nucleus, you have to deal with the coupling between the two odd spins.