My iMac is having increasing trouble connecting to the Internet.

My iMac (Yosemite 10.10.5) is having increasing trouble connecting to the Internet. Everything has been AOK for over a year with this same configuration, then it started out just taking a minute to connect whenever I opened mail or message or a browser. Now it will not connect except periodically.

This is a wired connection so it’s not a WiFi issue. Neither is it a DNS issue. The pipe is not to blame because I have hooked up numerous laptops to it and all is functioning well (3 up/30 down, just like it’s supposed to be). When it’s not connecting my network display in system prefs states “cable unplugged.” I have tried fiddling with the ethernet cable at the connector in the rear of the display to ascertain whether that is the cause, but I cannot get it to malfunction when I am connected.

The hard drive has ample room (573 gigs available), the activity monitor looks good (over 90% idle at the moment) and, anyway, I keep few programs open simultaneously and always store my big files on external hard drives. I run disc utility religiously so I don’t think it’s a file corruption. I’m wondering if it might be a hardware issue internally that is randomly toggling my online status off and on. The only other thing I can imagine is that there’s some sort of invasive file stuck in there somewhere that’s wreaking progressive havoc.

I am stumped. I would be grateful for any suggestions.