We report a novel mechanism of boundary-sensitive PT symmetry breaking in one-dimensional Floquet systems. By designing a time-periodic driving protocol, we realize a Floquet Hamiltonian that is Hermitian under periodic boundary conditions yet acquires non-Hermitian boundary terms under open boundary conditions due to the noncommutativity of the driving Hamiltonians. We show that boundary-induced PT breaking can occur when the boundary term couples Bloch modes that become degenerate after their quasienergy bands cross the frequency Brillouin zone boundary. For a single band, this requires the quasienergy bandwidth to span the entire frequency Brillouin zone, whereas in multi-band systems the relevant threshold is determined by the bands that can be coupled by the boundary terms. In the PT -broken phase, the eigenstates exhibit scale-free localization arising from the system-size scaling of the induced non-Hermitian terms. The associated wavepacket accumulation near the boundaries provides a dynamical signature of the transition. Finally, we present a general framework for constructing single- and multiband models exhibiting this phenomenon.