The influence of the stabilizing phospholipid-coating on the nonlinear dynamics of ultrasound contrast agent microbubbles is investigated. We record the radial dynamics of individual microbubbles with an ultra high-speed camera as a function of both driving pressure and frequency. The viscoelastic shell was found to enhance the nonlinear bubble response at acoustic pressures as low as 10 kPa. For increasing acoustic pressures a decrease of the frequency of maximum response was observed for a distinct class of bubbles, leading to a pronounced skewness of the resonance curve, which we show to be the origin of the "thresholding" behavior (Emmer et al. 2007). For the other bubbles, the frequency of maximum response was found to lie just above the resonance frequency of an uncoated microbubble and to be independent of the applied acoustic pressure. The shell-buckling bubble model (Marmottant et al. 2005), which accounts for buckling and rupture of the shell, captures both cases for a unique set of the shell parameters, the relevant parameter being the phospholipid concentration at the bubble interface.