Text message sent from my phone that I didn't send - This is possible, by three means (this is not exhaustive, there may be others):
"Inside job": the phone was compromised and a tool to remotely control it was installed on it and used to remotely send messages, without requiring nor notifying the user. This requires either a physical access to the phone or knowing the person's gmail account credentials (username + password and possibly phone text access if 2-steps authentication is enabled). It has always been possible to do so with Android (see AirDroid app), but it's becoming more and more difficult to remotely delete messages or bypass the Android default text manager (before Android KitKat, apps could delete messages without requiring user's consent, now it's impossible, at least theoretically - I'm sure a pro hacker can do something about that, but this requires a very high level of expertise, you cannot find such an exploit on the internet currently). This is the most probable scenario (if your client is not lying of course) if there are multiple messages, particularly over several days.
Buggy network routing: it can happen sometimes that one message gets lost in the network and that someone gets a message he shouldn't. This is very rare (I already experienced it myself), because there are error correcting codes that try hard to prevent any error in the network and in the messages, but this is not impossible, just highly unlikely. Since this is very unlikely to happen, if it does happen, only one or two messages will be wrongly misrouted to a person (not a string of messages and not over several days, it can only be punctual). Also, since this is because of a network error, the content will probably be totally unrelated to the implied persons (ie, content non pertinent, hour and time totally unrelated to the person's activity like during night, etc). There is a discussion about such an instance here. SMS Spoofing: Just like the buggy network routing, but done intentionally: any sender can manipulate the sending address of a SMS. This is done legitimately by some companies to replace the sender personal number by the one of the company or by alphanumeric text. Operators can usually detect when SMS is spoofed, so you may try to give a call to your client's operator to check if this was the case or not.
However, you didn't precise what kind of phone your client is using, nor the way the messages were sent (SMS, iMessage, WhatsApp, etc.) so the modality can change (but these techniques generally remain possible, although they may be more difficult with those other networks). Also, remember that SMS is generally always supported in all networks as an alternative to send and receive text messages, and I'm not sure these apps will tell you what kind of network was used for each message. For example, iMessage allows you to send SMS instead, and it streamlines all messages in one same interface.
