Periodic dips observed in \approx 20 % of low-mass X-ray binaries are thought to arise from obscuration of the neutron star by the outer edge of the accretion disk . We report the detection with the Rossi X-ray Timing Explorer of two dipping episodes in Aql X-1 , not previously a known dipper . The X-ray spectrum during the dips exhibited an elevated neutral column density , by a factor between 1 and almost two orders of magnitude . Dips were not observed in every cycle of the 18.95-hr orbit , so that the estimated frequency for these events is 0.10 _ { -0.05 } ^ { +0.07 } cycle ^ { -1 } . This is the first confirmed example of intermittent dipping in such a system . Assuming that the dips in Aql X-1 occur because the system inclination is intermediate between the non-dipping and dipping sources , implies a range of 72– 79 ^ { \circ } for the source . This result lends support for the presence of a massive ( > 2 M _ { \odot } ) neutron star in Aql X-1 , and further implies that \approx 30 additional LMXBs may have inclinations within this range , raising the possibility of intermittent dips in those systems also . Thus , we searched for dips from 24 other bursting systems , without success . For the system with the largest number of dip phases covered , 4U 1820 - 303 , the nondetection implies a 95 % upper limit to the dip frequency of 1.4 \times 10 ^ { -3 } cycle ^ { -1 } .