This paper proposes a simple adaptive sensing and group testing algorithm termed Compressive Adaptive Sense and Search (CASS). The algorithm is shown to be near-optimal in that it succeeds at the lowest possible signal-to-noise (SNR) levels. Like Compressed Sensing, the CASS algorithm requires only k log n measurements to recover a k-sparse signal of dimension n. However, CASS succeeds at SNR levels that are a factor log(n) less than required by standard Compressed Sensing. From the point of view of constructing and implementing the sensing operation as well as computing the reconstruction, the proposed algorithm is comparatively less computationally intensive than standard compressed sensing. CASS is also demonstrated to perform considerably better in simulation. To the best of our knowledge, this is the first demonstration of an adaptive sensing algorithm with near-optimal theoretical guarantees and excellent practical performance.