This issue occurs when mysms is not used as default SMS app on your smartphone (Android > 4.4). Sent SMS messages from a remote client are not really split up, they are just displayed as separate messages on your end. That means the recipient will still receive the text/content in one long message.

Due to some restrictions of the Android operating system, mysms is not able to write long messages into the SMS database if mysms is not the default texting app. As a result, mysms is saving the messages as single ones, even though they are sent out correctly. If mysms is used as default app, messages are displayed correctly.