Are you able to identify any patterns to the problem? For example, do you have any scheduled backups? Does the problem happen while those are running?
Is FishEye trying to index a huge changeset?
Do you have any huge Crucible review?
Do you sync users with an external tool like JIRA / Crowd / LDAP? How many users are synced, and how often?
Has your instance always behaved this way, or has this problem started to happen lately? If the latter, are you aware of any changes in instance settings or even server settings?
Your instance version is rather old, and many performance improvements have been made since then. Can you give the latest FishEye version a try, to see if it performs better?
What are the current JVM settings? Do you see any OutOfMemory issues?
You might want to raise a support issue, so that the support team can have a look at the instance logs and configuration.
If this answer was helpful, please allow me to ask you to mark my answer as accepted in order to have it in the top of the thread, also helping other customers.
Thanks in advance!