Not if they upgraded both their hardware and software along the way. Even without upgrades, their computer might function just fine, but they should not expect to browse websites created with technologies that were not yet dreamed of when their computer was born.
People expect to make phone calls on their 5 year old phones, but it would be unreasonable for them to complain that they cannot download the latest version of snapchat on it. People with old browsers and hardware can expect old websites and software to work for them. It is unreasonable for developers to be required to create new websites that work for outdated and insecure browsers.
The "upgrades" necessary would be a new computer in all but case, so unless your name is Theseus, you've got a new computer.
Or Trigger...
http://www.frequency.com/video/only-fools-horses-truth-behin...
The comparison doesn't hold though.