Administering incomplete data communications messages in a parallel computer that includes a plurality of compute nodes, with each compute node including a processor and a messaging accelerator, includes: transmitting, by a source messaging accelerator to a destination messaging accelerator, a message, including processing a messaging descriptor describing the message and setting, in the message descriptor, a flag indicating the message has been sent; transmitting, by the source messaging accelerator to a destination messaging accelerator responsive to processing an acknowledgement request descriptor corresponding to the message, a request for acknowledgment of receipt of the message; receiving, by the source messaging accelerator from the destination messaging accelerator, a negative acknowledgment (NACK) indicating that the message was not received at the destination messaging accelerator; and clearing, by the source messaging accelerator in the message descriptor, the flag indicating that message has been sent.