The associated network connection failed. That's all that error means. Why... is an open question, and depends on what tool returned that error, the state of the network, and which particular socket communications operation(s) failed.
Use Disk Utility, get a complete disk backup to an external disk — old hardware can fail, corruptions and weird behavior can be a nascent hardware error, and the more time spent trying to "fix" the error can be a trade-off against the shortening life of a failing disk. Run the hardware diagnostics.
For slow network activity, I'd also verify that local DNS is configured and operating, as well — within a NAT'd network, that cannot be the ISP DNS servers, it has to be the DNS running on the local server. The command-line command sudo changeip -checkhostname should return some info on your local network and an indication that no changes are required with the following footer after the config data:
The names match. There is nothing to change.
dirserv:success = "success"
If it returns errors, then resolve those.
If generic disk tools — tools that are not using the network — are running slowly, then the I/O path or the disk storage may well be failing. Boot a copy of OS X from a key disk, and use Disk Utility get a disk backup to external storage. If that backup is slow, then it's probably failing storage. On a cheese-grater Mac Pro, the disks are easy to swap. While you have the box open, clean out the accumulated dust, as that can cause the system to overhead and get flaky.
If there's nothing obvious, I'd probably then look at reinstalling from distro, and migrating in the data.