"Average case performance" is an oft-cited motivation for self-timed design. In self-timed designs, computations proceed according to handshakes, and these handshakes can reflect the actual time required for operations rather than the worst-case time. The intuitive argument is that this should lead to systems whose performance reflects the average-case performance of their components. This paper shows that such intuition is often wrong. This paper describes a connection between self-timed circuits and percolation networks. Percolation networks are a class of infinite graphs originally used to model critical phenomena arising from fluid flows in porous media. This paper shows how these techniques can be used to show the frequent existence of long chains of slow operations in self-timed designs. These chains can give rise to performance that is closer to worst-case than average-case. This paper makes three contributions. First, it describes a fundamental connection between percolation networks and self-timed circuits. Second, it presents novel methods for studying percolation networks that arise in the analysis of self-timed circuits. Third, it gives examples of self-timed circuits whose performance is limited by percolation phenomena.