We here introduce the concept of fragile dislocation modes, which are localized only in a fraction of a topological phase while otherwise leaking into the bulk continuum. As we demonstrate here, such dislocation modes are hosted in an obstructed atomic topological phase in the two-dimensional Su-Schrieffer-Heeger model but only in a finite region with an indirect gap at high energy. They are realized as chiral pairs at finite energies with protection stemming from a combination of the chiral (unitary particle -hole) and the point group ( C- 4 v ) symmetries, but only when the indirect gap is open. In this regime, we corroborate the stability of the defect modes by following their localization and also by explicitly adding a weak chemical potential disorder. Our findings should, therefore, be consequential for the experimental observation of such modes in designer topological crystals and classical metamaterials.