It depends on what the text field is supposed to accept and what it's for.

If it's meant to accept specific formats of data, your tests will be different than if it's meant to accept free-form data. For example, text fields are often used for postal code entry. For that, there's likely to be validation that the postal code format matches the official format for the country (USA: xxxxx or xxxxx-xxxx where x is any number, Australia: xxxx where x is any number - those are the only two I know offhand).

Some other considerations:

Is there a way to get invalid input into the field? Most browsers will allow drag and drop or copy/paste into fields: this is a common way to put invalid data into a field. The key thing is to handle the invalid input gracefully.

Is there a way to override the field length limitations? Copy/paste and drag and drop can do this. I've also seen holding down a key do it, particularly if a control key is involved.

Are there limits on the characters the field should accept? If you're limiting to English alphanumeric, the field needs to handle other characters gracefully.

Special character handling - does the field handle characters that have specific meanings in HTML, JavaScript, the back-end programming language used, SQL? Does it handle leading spaces? Should it handle leading spaces?