Why worry about checking the tailpiece and fine tuners?
Fine tuners often loosen and may cause buzzes, or wear the strings so they break prematurely. It may be necessary to file down the fine tuner slot to prevent wear on the strings. Some strings, especially Dominants, are larger in diameter than the fine tuners are designed for, and the fine tuners may have been bent or filed to accommodate them. Fine tuners that are built into the tailpiece, or sit on top of the tailpiece, are superior to the ones that extend out from under the tailpiece, because they preserve the ratio of string length to afterlength.
The afterlength of a string affects its tone. In most cases, the tailgut should be shortened so the tailpiece will be very close to the saddle. Check the condition of the tailgut also. Modern tailgut is synthetic and does not have the wear problems associated with old-fashioned gut, but the ends may have been left long and may vibrate or cause buzzes.