The recent detection of a 3-hr X-ray flare by the Chandra Observatory has raised the possibility of enhanced emission over a broad range of wavelengths from Sgr A*, the suspected 2.6 x 106 solar mass black hole at the Galactic Center, during a flaring event. We have, therefore, reconstructed 3-hr data sets from 2 micron speckle and adaptive optics images ( theta _core = 50 - 100 mas) obtained with the W. M. Keck 10-m telescopes between 1995 and 2001. In 25 separate observations, no evidence of any significant excess emission associated with Sgr A* was detected. The lowest of our detection limits gives an observed limit for the quiescent state of Sgr A* of 0.09+/-0.005 mJy, or, equivalently, a dereddened value of 2.0+/-0.1 mJy, which is a factor of 2 lower than the best previously published quiescent value. Under the assumption that there are random 3-hr flares producing both enhanced X-ray and near-infrared emission, our highest limit constrains the variable state of Sgr A* to <=0.8 mJy (observed) or 19 mJy (dereddened). These results suggest that the model favored by Markoff et al. (2001), in which the flare is produced through local heating of relativistic particles surrounding Sgr A* (e.g., a sudden magnetic reconnection event), is unlikely, because it predicts peak 2 micron emission of 300 mJy, well above our detection limit.