### Check for existing issues

- [x] I have searched the existing issues and checked that my issue is not a duplicate.

### What happened?

We run LiteLLM proxy with PostgreSQL as the database backend and multiple workers.

After the service had been running for some time under traffic, all normal user API keys started failing for every model call with:

```text openai.AuthenticationError: Error code: 401 - { "error": { "message": "Authentication Error, All connection attempts failed", "type": "auth_error", "code": "401" } }

### Steps to Reproduce

```markdown 1. Deploy LiteLLM proxy with PostgreSQL backend. 2. Configure PostgreSQL with `max_connections=300`. 3. Run LiteLLM with multiple workers, e.g. `--num_workers=8`. 4. Enable normal user API keys stored in PostgreSQL. 5. Generate concurrent chat completion traffic using normal user keys. 6. Run additional background services that also connect to the same DB, such as usage sync / log archive / custom quota callbacks. 7. Once PostgreSQL reaches `max_connections`, call:

```bash curl https://<proxy-host>/v1/chat/completions \ -H "Authorization: Bearer <normal-user-key>" \ -H "Content-Type: application/json" \ …