Variational quantum algorithms dominate gate-based applications of modern quantum processors. The so-called layerwise trainability conjecture appears in various works throughout the variational quantum computing literature. The conjecture asserts that a quantum circuit can be trained piecewise, e.g., that a few layers can be trained in sequence to minimize an objective function. Here, we prove this conjecture false. Counterexamples are found by considering objective functions that are exponentially close (in the number of qubits) to the identity matrix. In the finite setting, we found abrupt transitions in the ability of quantum circuits to be trained to minimize these objective functions. Specifically, we found that below a critical (target-gate-dependent) threshold, circuit training terminates close to the identity and remains near to the identity for subsequently added blocks trained piecewise. A critical layer depth will abruptly train arbitrarily close to the target, thereby minimizing the objective function. These findings shed light on the divide-and-conquer trainability of variational quantum circuits and apply to a wide collection of contemporary literature.