Isomorphism and stable isomorphism in "real" and "quaternionic" K-theory

We find lower bounds on the rank of a "real" vector bundle over an involutive space, such that "real" vector bundles of higher rank have a trivial summand and such that a stable isomorphism for such bundles implies ordinary isomorphism. We prove similar lower bounds also for "quaternionic" bundles. These estimates have consequences for the classification of topological insulators with time-reversal symmetry.