The overflow property is the way to go. In particular, if you want the extra text to be ignored, you can use "overflow:hidden" as a css property on the text.
In general, when a browser has an unbreakable object, such as a long string without spaces, it can have a conflict between various size constraints - those of the string (long) vs its container (short). If you see different behavior in different browsers, they are just resolving this conflict differently.
By the way, there is a nice trick available for long strings - the <wbr> tag. If your browser sees longstringlongstring, then it will try to fit it in the container as a single, unbroken string -- but if it can't fit, it will break that string in half at the wbr. It's basically a break point with a implicit request to not break there, if possible (sort of like a hyphen in printed texts). By the way, it's a little buggy in some versions of Safari and Opera - check out this quirksmode page for more.