An in-depth analysis of nonautonomous bifurcations of saddle-node type for scalar differential equations x′=−x2+q(t)x+p(t), where q:R→Randp:R→Rare bounded and uniformly continuous, is fundamental to explain the absence or occurrence of rate-induced tipping for the differential equation y′= (y−(2/π) arctan(ct))2+p(t) as the rate c varies on [0,∞). A classical attractor-repeller pair, whose existence for c= 0 is assumed, may persist for any c >0, or disappear for a certain critical rate c=c0, giving rise to rate-induced tipping. A suitable example demonstrates that one can have more than one critical rate, and the existence of the classical attractor-repeller pair may return when c increases.