Network latency test - undetermined
Reason unknown
Network route test - failed
Reason unknown
My guess is these errors will give you a clue, and as Robert mentioned, home networks have too many variables for a forum like this to probably figure out.
With that said, I know I have to replace my modem (wireless and wired) every year or so. They just start having flaky connection issues. In on sense, this seems like it would be too often, but a technician put it in perspective for me once. Considering most home networks are on 24-7, and these things aren't build with the highest quality parts most the time, the heat from being on all the time can make them act up. Sometimes cheaper to just get a new one that chase down why.
If not a simple piece of hardware like that, then your folks might have some firewall or other administration in place. And again, way too many possibilities. Maybe your folks will give you the admin password so you can check it out.
Also, are you sure your student license is still valid outside of school?
Some suggestions.....wait a day and see if its temporary; bring your machine to a friends house or another location and see if you can connect to ArcGIS Online fine there (I'm assuming you tried connecting thru the browser only too....ruling out that GIS software from getting in the way), start looking at the network.