We examine the budding of a nanoscale particle through a lipid bilayer using molecular dynamics simulations, free energy calculations, and an elastic theory, with the aim of determining the extent to which equilibrium elasticity theory can describe the factors that control the mechanism and efficiency of budding. The particle is a smooth sphere which experiences attractive interactions to the lipid head groups. Depending on the parameters, we observe four classes of dynamical trajectories: particle adhesion to the membrane, stalled partially wrapped states, budding followed by scission, and membrane rupture. In most regions of parameter space we find that the elastic theory agrees nearly quantitatively with the simulated phase behavior as a function of adhesion strength, membrane bending rigidity, and particle radius. However, at parameter values near the transition between particle adhesion and budding, we observe long-lived partially wrapped states which are not captured by existing elastic theories. These states could constrain the accessible system parameters for those enveloped viruses or drug delivery vehicles which rely on exo- or endocytosis for membrane transport.