Websites are Down, "too many connections error" disk space reached?

We have a dedicated server and the websites are currently down. Our web host says they cannot help with it, but says that the /var partition is full and that we need to re-configure the server partitions. I get an error "too many connections" before rebooting the server, but now nothing comes up. We need help immediately. Thanks.

Kemahiran: Linux, Sistem Admin

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ID Projek: #4108127