We present results from the largest systematic investigation of broad absorption line (BAL) acceleration to date. We use spectra of 140 quasars from three Sloan Digital Sky Survey programs to search for global velocity offsets in BALs over timescales of approximate to 2.5-5.5 years in the quasar rest frame. We carefully select acceleration candidates by requiring monolithic velocity shifts over the entire BAL trough, avoiding BALs with velocity shifts that might be caused by profile variability. The C IV BALs of two quasars show velocity shifts consistent with the expected signatures of BAL acceleration, and the BAL of one quasar shows a velocity-shift signature of deceleration. In our two acceleration candidates, we see evidence that the magnitude of the acceleration is not constant over time; the magnitudes of the change in acceleration for both acceleration candidates are difficult to produce with a standard disk-wind model or via geometric projection effects. We measure upper limits to acceleration and deceleration for 76 additional BAL troughs and find that the majority of BALs are stable to within about 3% of their mean velocities. The lack of widespread acceleration/deceleration could indicate that the gas producing most BALs is located at large radii from the central black hole and/or is not currently strongly interacting with ambient material within the host galaxy along our line of sight.