1. A text messaging system made up of a plurality of information terminals provided with communications means, and servers connected to the information terminals via communications lines responsive to the communications means, the text messaging system comprising: transmission means provided in the information terminals for transmitting to the servers text messages addressed to other of the information terminals; reception means provided in the information terminals for receiving from the servers text messages from other of the information terminals; status detection means provided in the information terminals for detecting and reporting to the servers operational status including at least status of connection with the communications lines; reception means provided in the servers for receiving text messages from the information terminals; transmission means provided in the servers for transmitting text messages received from ones of the information terminals to others of the information terminals; status administration means provided in the servers for holding per user information-terminal operational status reported from said status detection means; and decision means provided in the servers for deciding transmission modes for the text messages according to the information terminal operational status.