Implementations of the Transport Layer Security (TLS) protocol must handle a variety of protocol versions and extensions, authentication modes and key exchange methods, where each combination may prescribe a different message sequence between the client and the server. We address the problem of designing a robust composite state machine that can correctly multiplex between these different protocol modes. We systematically test popular open-source TLS implementations for state machine bugs and discover several critical security vulnerabilities that have lain hidden in these libraries for years (they are now in the process of being patched). We argue that these vulnerabilities stem from incorrect compositions of individually correct state machines. We present the first verified implementation of a composite TLS state machine in C that can be embedded into OpenSSL and accounts for all its supported ciphersuites. Our attacks expose the need for the formal verification of core components in cryptographic protocol libraries; our implementation demonstrates that such mechanized proofs are within reach, even for mainstream TLS implementations.