I have a couple of jobs that use a shared resource (database), which sometimes can cause builds to fail in the (rare) event that the jobs happen to get triggered simultaneou
That's an old question, but the topic can still be relevant, especially when running application tests on Jenkins.
The Lockable Resources Plugin allows you to define lockable resources that can be used by builds. If your build requires an resource, it takes the lock. If a second build requires the same resource (which then is already locked), it will be queued for the resource to be free.
Although the docs use computers or printers as examples for lockable resources, the database example from above should work as well.
In opposite to the Locks and Latches Plugin mentioned in answers from 2012, this package seems to be currently maintained (currently ~2016).