This is definitely possible, I have done it many times - it just warns you that on saving you can't go back to 6.1.
Does the same workbook open in 6.1? It sounds like the extract file was moved or deleted.
Certainly a 6.1 .twb can be opened in 7.0. Are you opening a .twb or .twbx ? If the latter, then that could be a problem since the Extract formats are different. If the former, then as Dimitri suggested it is probably a file location problem and you should verify that the actual .tde file is where the .twb expects it to be.
You can have both 6.1 Desktop and 7.0 Desktop installed on the same machine; just don't ever run both simultaneously (Licensing violation). If the .twb works fine in 6.1 and breaks in 7.0, then something else is afoul and you may need Support for that.