Urgh..
OK, first off, make sure that it's an actual "ISP change", as there might be occassions when someone resells the connection. For example, my home Internet is through Earthlink, but TimeWarner Cable provides the physical connection from my house back to the local office. So, were I to have a problem with Earthlink & decide to switch to TimeWarner, it's quite possible that any equipment issue on "their" end will remain, because I'm still on the same equipment.
Next, make sure that it's not some independent ISP that depends on a bigger company where it "co-locates" its equipment. For example, back in the dialup days (when pretty much copper lines were the only type of non-cellular phones), the former "Baby Bell" companies -- Pacific Bell, Southwestern Bell, Ameritech, etc. -- were required by law to allow small, independent telcos to have access to the local central offices for the "last mile" portion...which meant that, if the big telco's equipment had issues, it could affect the independent telco's service as well.
This could also happen if multiple ISPs are connecting into the same main Internet backbone, & the backbone has equipment issues. I remember back in the day (pre-9/11) when all of the major long-distance telcos (Sprint, AT&T, Qwest Communications, MCI, etc.) had major customer order completion issues in NYC because there was a major OC fiber ring that hadn't been completed yet (was something like months, getting close to a year, behind schedule), & everyone was scrambling to find alternate routes to get customer orders completed so they could start collecting the monthly bills. Obviously, once that ring went up, every company hooked into it would be dependent upon the common equipment, & therefore any equipment failures or issues could affect not only multiple customers but also multiple providers.