The existence of nonlinear Newton-Wigner delay-times in the femtosecond range is demonstrated using a pulsed laser beam with rotating polarization impinging on an interface together with a time-resolved differential detection at a "magic angle". Experimentally, increasing the incident power by a factor of 10 leads to a decrease in the delay-time by more than 70% at total reflection. In contrast, below the critical incidence angle, the same power variation leads to a huge increase in the delay-time. Intrinsic nonlinear Newton-Wigner delays are also expected in quantum reflection and the magic angle method could be extended to shorter pulses, down to the attosecond domain. Possible applications are discussed.