I understand it to mean a test system that is updated each night.
No, a nightly build is not that. It is an automated compilation of your source code, normally accompanied by automatic running of unit and integration tests (though strictly speaking, the compilation part is the build).
Is it excusable to say that the system I develop is too complex for a nightly build?
Excusable? It is an excuse. But you don't have to start with 15 databases being covered by tests. You can start by getting an automated compilation. Then you can add unit tests (that don't touch the database). Then start with integration tests with one database, followed by another, going on till you covered all 15 and the important functionality.
In general, the above can describe a Continuous Integration (CI) environment, where the difference is that in most CI environments a build is triggered on every commit/checkin/push and tests are run (in some places, if all tests are green, the product is also automatically deployed).