We used near-infrared data obtained with the Wide Field Camera 3 on the Hubble Space Telescope to identify objects having the colors of brown dwarfs (BDs) in the field of the massive galactic cluster NGC 3603. These are identified through a combination of narrow-and medium-band filters which span the J and H bands and are particularly sensitive to the presence of the 1.3-1.5 mu m H2O molecular band unique to BDs. We provide a calibration of the relationship between effective temperature and color for both field stars and BDs. This photometric method provides effective temperatures for BDs to an accuracy of +/-350K relative to spectroscopic techniques. This accuracy is shown to be not significantly affected by either stellar surface gravity or uncertainties in the interstellar extinction. We identify nine objects having effective temperatures between 1700 and 2200 K, typical of BDs, observed J-band magnitudes in the range 19.5-21.5, and that are strongly clustered toward the luminous core of NGC 3603. However, if these are located at the distance of the cluster, they are far too luminous to be normal BDs. We argue that it is unlikely that these objects are either artifacts of our data set, normal field BDs/M-type giants, or extragalactic contaminants and, therefore, might represent a new class of stars having the effective temperatures of BDs but with luminosities of more massive stars. We explore the interesting scenario in which these objects would be normal stars that have recently tidally ingested a hot Jupiter, the remnants of which are providing a short-lived extended photosphere to the central star. In this case, we would expect them to show the signature of fast rotation.