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Using magnetically trapped atomic hydrogen as an example, we investigate the prospects of achieving Bose-Einstein condensation in a dilute Bose gas. We show that, if the gas is quenched sufficiently far into the critical region of the phase transition, the typical time scale for the nucleation of the condensate density is short and of $O(\hbar/k_{B}T_{c})$. As a result we find that thermalizing elastic collisions act as a bottleneck for the formation of the condensate. In the case of doubly-polarized atomic hydrogen these occur much more frequently than the inelastic collisions leading to decay and we are lead to the conclusion that Bose-Einstein condensation can indeed be achieved within the lifetime of the gas.