A data processing apparatus includes error detection and correction circuitry with an associated hard-error memory buffer. When a correctable hard-error is detected associated with a memory access to a memory, if the hard-error memory buffer is already full, then this correctable hard-error is escalated to be handled as an uncorrectable hard-error. The escalated uncorrectable hard-error is then handled by uncorrectable error handling circuitry (fatal error circuitry) which may trigger an abort of corresponding processing operations by a processor core and force the relinquishing of resources within other circuit elements such as a store buffer.