There shouldn't be a text message limit, at least nothing that isn't the same on all devices. Text messages (actually they are SMS messages) are limited to 160 characters, no matter what the device.It is just that some (many) devices will let you keep typing on for a while, and then send them out as individual 160 max character messages. This is also why when you sent a message to an email address from a text (SMS) message, and it is longer than that, the message may never show up in the email, or only the first part/message will show. And that may actually be shorter than the 160 characters, as it can include other info when sent to email. Also, if you send a message to a phone/text from email, that is limited even more due to mail headers, subject lines, etc.
There is no way to increase that 160 character limit per message. Then, on top of this, you also have the phones limitation on how many characters it will allow to be typed before it stops. Which is not related to the per message character limit, but will be then sent out as multiple texts.
To avoid this, in most cases that is, you can send the message not as a text (SMS) message but as a MMS (multimedia) message. Just include an image of some type. Can be a tiny one even. That will change the message type and allow for longer messages. Just remember though, the person you are sending to must also be able to get MMS (Multimedia) messages.
You will have to check your phone's manual or with the manufacturer, to see what the limitation of max characters for your particular device is. However, again, there is no way to increase that.