The major obstacle to the direct detection of companions to nearby stars is the overwhelming brightness of the host star. Current instruments employing the combination of adaptive optics (AO) and coronagraphy can typically detect objects within 2''of the star that are~ 10 4-10 5 times fainter. Correlated speckle noise is one of the biggest obstacles limiting such high-contrast imaging. We have obtained a series of 284 8 s, AO-corrected, coronagraphically occulted H-band images of the star Vega at the 3.63 m AEOS telescope located on Haleakala, Hawaii. This data set is unique for studying the temporal behavior of speckle noise and represents the first time such a study on highly corrected coronagraphic AO images has been carried out in a quantitative way. We find the speckle pattern to be highly stable in both position and time in our data. This is due to the fact that the AO system corrects disturbances to the stellar wave front at the level where the instrumental wave front errors dominate. Because of this, we find that our detection limit is not significantly improved simply with increased exposure time alone. However, we are able to improve our dynamic range by 1.5-2 mag through subtraction of static/quasi-static speckles in two rotating frames: the telescope pupil frame and the deformable mirror frame. The highly stable nature of speckles will exist for any program using coronagraphy and high-order AO. Furthermore, from our data, we are able to constrain the mass of any purported companion to Vega to be less than~ 45M J at 8 AU and less than~ 30M J at 16 AU, radii not previously probed at these sensitivities.