I am curious as to what makes a database open. Specifically, on another SE site I have been in discussions with a developer at Mendeley (a reference manager and social network) about whether it qualifies as FOSS. His contention is that Mendeley is FOSS since the data are made available via an API. I think it is unreasonable to label software that doesn't release the source code as FOSS, but I am not sure if the API based access to database is enough to label it as open data. This might be related to Open database APIs for journal article metadata, but I did not see Mendeley mentioned. I cannot find any licensing information on the site and they talk about authorization and registration which don't sound like open terms to me.
Mendeley does not seem to grant me any license to reuse the content (in particular the academic papers uploaded by other users), so by default the data must be considered closed: We are not allowed to redistribute it. Even if an API allows us to retrieve it:
You may not use our Services to [...] download, use or re-use any Academic Papers without authorization.
Mendeley's software is clearly not FOSS, as seen in the Terms:
you undertake not to copy, rent, lease, sub-license, loan, translate, merge, adapt, vary or modify the whole or any part of our Software
The question about the data itself is independent of the software itself being open source or not. You can run MediaWiki (open source software) to run a private server and store non-open data on it. On the opposite, you could run IIS (proprietary software) to host open data.