The mechanisms underlying cardiac fibrillation have been investigated for over a century, but we are still finding surprising results that change our view of this phenomenon. The present study focuses on the transition from normal rhythm to atrial fibrillation associated with a gradual increase in the pacing rate. While some of our findings are consistent with existing experimental, numerical, and theoretical studies of this problem, one result appears to contradict the accepted picture. Specifically we show that, in a two-dimensional model of paced homogeneous atrial tissue, transition from discordant alternans to conduction block, wave breakup, reentry, and spiral wave chaos is associated with transient growth of finite amplitude disturbances rather than a conventional instability. It is mathematically very similar to subcritical, or bypass, transition from laminar fluid flow to turbulence, which allows many of the tools developed in the context of fluid turbulence to be used for improving our understanding of cardiac arrhythmias.