Open CDR in Illustrator

We have just upgraded our office graphics computer. As part of that, the company has bought Illustrator, (despite our objections) because they said it's more industry standard.
However, we have used Corel Draw draw 13 years and every publication figure we have is a CDR file.
Shouldn't be a problem because  -- according to Adobe -- Illustrator can open Corel draw files from versions 5 to 10. Right?
Except it can't. It says they're unrecognised files.
Open source programmes and online converters are not permitted (because we're part of a local government department.)
So can someone please tell me how to get this software to do what the company claims it does?
There are a LOT of files involved and converting them all in Corel is not an option.
 
If Illustrator actually cannot do this, why does Adobe say that it does?