We find exact localized time-periodic solutions with frequencies inside the linearized spectrum [intraband discrete breathers (IDBs)] in random nonlinear models using a new self-consistent method. The IDB frequencies belong to intervals between forbidden gaps generated by resonances with the linear modes, becoming fat Cantor sets in infinite systems. When localized IDBs are continued versus frequency, they delocalize and become multisite IDBs (not predicted by existing theorems), which can propagate energy. Some implications for energy relaxation in glasses are discussed.