The widely used Nose-Hoover chain (NHC) thermostat in molecular dynamics simulations is generally believed to impart the canonical distribution as well as quasi- (i.e., space-filling) ergodicity on the thermostatted physical system (PS). Working with the standard single harmonic oscillator, we prove analytically that the two-chain Nose-Hoover thermostat with unequal thermostat masses approaches the standard Nose-Hoover dynamics, and hence the PS loses its canonical and quasiergodic nature. We also show through numerical simulations over substantially long times that for certain Poincaré sections, for both the equal and unequal thermostat mass cases, the bivariate distribution function of position and momentum (x,p) and of reservoir degrees of freedom (ξ,η) lose their Gaussian nature. Further, the four-dimensional x-p-ξ-η extended phase space exhibits two holes of nonzero measure. The NHC thermostat therefore does not generate the canonical distribution or preserve quasiergodicity for the PS.