If this persists, return the Office package and buy from a more reputable source. This is a case where I would recommend a 3rd party app uninstaller like App zapper to get all the bits of office off the OPs computer and then reinstall from the disc. A Mac computer that has an Intel processor Mac OS X version 10.5.8 (or a later version) 1 gigabyte (GB) or more of memory (RAM) 2. I find it odd that it never asked for a new key though. Microsoft support said that support for MS Office 2011 has not ended. In the case of the OP, it sounds like the product key from the purchased version may have been stolen or otherwise compromised and that an MS product using that key is being used on the LAN. When the other user closes their MS App, you can then open it.

If you have a single copy and and have multiple users utilizing it (with the same registration key) on a LAN, when a second user attempts to open an Office App, you will be told that another user is using a MS Product with that Identification key and that you can't open any Office Apps. Office Mac 20 are interesting because unlike other versions, they check their local LAN for other users using the same key. This was even true when multiple licenses were obtained and the package was installed on office computers on a LAN. Every version of Mac Office I've ever used (Since Mac Office 4) has asked for a key when any of the apps was opened for the first time.