Photometric observations in the R and I bands of the detached M-type double-lined eclipsing binary CU Cnc have been acquired and analysed. The photometric elements obtained from the analysis of the light curves have been combined with an existing spectroscopic solution to yield high-precision (errors<2%) absolute dimensions: M_A=0.4333+/-0.0017 Mo, M_B=0.3980+/-0.0014 Mo, R_A=0.4317+/-0.0052 Ro, and R_B=0.3908+/-0.0094 Ro. The mean effective temperature of the system has been estimated to be Teff=3140+/-150 K by comparing multi-band photometry with synthetic colors computed from model atmospheres. Additionally, we have been able to obtain an estimate for the age (~320 Myr) and chemical composition ([Fe/H]~0.0) of the binary system through its membership of the Castor moving group. With all these observational constraints, we have carried out a critical test of recent stellar models for low-mass stars. The comparison reveals that most evolutionary models underestimate the radius of the stars by as much as 10%, thus confirming the trend observed by Torres & Ribas (2002) for YY Gem and V818 Tau. In the mass-absolute magnitude diagram, CU Cnc is observed to be dimmer than other stars of the same mass. After ruling out a number of different scenarios, the apparent faintness of CU Cnc can be explained if its components are some 10% cooler than similar-mass stars or if there is some source of circumstellar dust absorption. The latter could be a tantalizing indirect evidence for a coplanar (Vega-like) dusty disk around this relatively young M-type binary.