Border around a text box

Is it possible to have a border around a text box without have to make a separate rectangle?

No, you have to draw a separate rectangle and put it behind the text.

If we had 'text frames' we'd just have to provide all the same options as the rectangle tool has (e.g. rounded corners etc), it's just a lot simpler and more flexible all round that we do it this way.

Group or Soft Group them and they will act as one object.