Ground-based optical searches for faint stellar or planetary companions about other stars may be limited by speckle noise, which is the rapid intensity fluctuations that are due to motions of remnant atmospheric speckles. Adaptive optics (AO) can reduce residual wave-front phase errors to low values, substantially reducing the unwanted power in the speckle halo. At high correction, however, the noise in the halo will be dominated by anomalously bright "pinned" speckles that have a number of unusual properties. They can have negative intensities and will appear in spatially antisymmetric patterns; they are spatially pinned to Airy rings and have zero mean in a sufficiently long integration. Some of these properties may be used to reduce the unanticipated effect of pinned speckles on companion searches, depending on details of the AO system. But, in short exposures, pinned speckles dominate speckle noise over much of the inner halo for Strehl ratios S as low as 0.6 and over much of the outer halo too as Strehl and deformable-mirror actuator densities increase. I show that these anomalously bright pinned speckles are not included in the traditional expression for speckle power in an image, (1 - S), on which sensitivity estimates of future high-performance AO systems have been based.