$Q^2 \leq M^2$
in some natural units. It is absolutely impossible to violate this bound. However, general relativity implies a stronger bound that also includes the angular momentum $a$ in addition to the charge $Q$:
$Q^2 + a^2 \leq M^2$
One can also calculate the efficiency of energy transfer from accreting matter to radiation. She gets 6% for the Schwarzschild black hole, 42% for an extremal Kerr black hole, and Gimon & Hořava argue that string theory can get up to the 100% efficiency with configurations that obey the BPS bound but not the stronger black hole bound. These configurations would surpass the general relativistic Kerr limitations. They encourage astronomers to look for such efficient objects and realize a new way to prove string theory.
My personal guess based on our work on the weak gravity conjecture is that the black hole bound is also satisfied in string theory for localized macroscopic objects, up to small corrections. This belief of mine is supported by the observation that Gimon & Hořava don't have any explicit solution for their "superspinar". Of course, superspinars may fail to be associated with a classical solution but I personally find such a belief unnatural: a classical approximation should become valid for any kind of a large enough object.
And that's the memo.
Update: Later I realized that when we relax the black hole condition, we can easily find systems with greater angular momentum than allowed by the mass - for example the Solar System where the distant planets have "J" increasing with the radius "r".