  We compute the genus zero bridge numbers and give lower bounds on the genus
one bridge numbers for a large class of sufficiently generic hyperbolic twisted
torus knots. As a result, the bridge spectra of these knots have two gaps which
can be chosen to be arbitrarily large, providing the first known examples of
hyperbolic knots exhibiting this property. In addition, we show that there are
Berge and Dean knots with arbitrarily large genus one bridge numbers, and as a
result, we give solutions to problems of Eudave-Mu\~noz concerning tunnel
number one knots.
