One of the largest off-resonant nonlinear effects attributed to a third order nonlinear process has been reported in a DAN single crystal core fiber. Such fiber, designed for efficient second harmonic generation, showed large spectral broadening effects when femtosecond pulses around 625 nm were launched into it. Yamashita et al. compared the observed spectral broadening to that of a standard fused silica fiber and reported an intensity dependent refractive index 17000 times larger than that of fused silica. In this paper we identify the relevant physics of the large nonlinearities observed in the fibers. We have performed nonlinear interferometry measurements on a similar DAN fiber at 1.320 μm, obtaining a comparable magnitude, n2 = -0.8 × 10-3 cm2/GW, with a surprising negative sign. Furthermore, by a modification of the fiber cladding refractive index, it was possible to make second harmonic generation either allowed or forbidden. In our case, a nonlinear phase shift was only present when a second harmonic was also observed. In order to verify a relation between the large nonlinearity and the presence of the second harmonic generation, we performed z-scan measurements on DAN single crystals at 1064 nm. These results, in magnitude and sign, compared favorably with our interferometric measurements at 1319 nm.