...a little bit. Most people probably have no idea what version their browser is running these days. They tend to look after themselves; automatically applying updates as necessary. Both Chrome and Firefox are approaching a big milestone that may cause problems for some websites.
People that remember the Y2K bug may be ahead of us here, but the potential issue that's been discovered is that once these browsers hit version 100, some websites won't be able to cope with the 3 digit number, and will assume that the user is running version 00. If the content of that site is expecting the version number to be between a certain range and isn't accounting for "version 0", it may break in unexpected ways.
Mozilla tested this in advance and confirmed this hypothetical issue was genuine, and that it did cause a problem with a small number of sites in their testing, including HBO Go and Yahoo.
With this issue flagged up, web developers will hopefully now be checking their code to ensure their websites won't collapse when presented with 3 digit version numbers. Browser manufacturers are also keeping an eye on this and are ready to hack in some workarounds if necessary. We'll find out whether this has been successful on March 29th for Chrome and May 3rd for Firefox, when their respective version 100s are scheduled for release.