This problem is one that comes up quite a bit, BIRDS! Birds build nests in the gutters and they clog. If clogging was not enough, the constant noise from the birds is a nuisance. The gutters were also undersized for the size of the home and downspout positions.
We removed the old gutter system and carefully removed the nests. The nests were placed nearby and no birds were hurt during the installation of gutters. New gutters 6" and 3x4 downspouts were installed. We were even able to remove a downspout that was no longer needed due to water distribution to a larger gutter/downspout. MasterShield was then put on the system to prevent the leaves, shingle grit, and birds from getting into the gutters again. We now expect to get a call from their neighbors to deal with their bird problem.