Higher-order topological insulators are a recently discovered class of materials that can possess zero-dimensional localized states regardless of the dimension of the system. Here, we experimentally demonstrate that the topological corner-localized modes of higher-order topological systems can be symmetry-protected bound states in the continuum; these states do not hybridize with the surrounding bulk states of the lattice even in the absence of a bulk band gap. This observation expands the scope of bulk-boundary correspondence by showing that protected boundary-localized states can be found within topological bands, in addition to being found in between them.