As Joeqwerty has said in his comment, you need to expand your understanding of DNS. The reason why DNS changes take a while to propagate is down to recursive caching and Time To Live (TTL).
Consider this scenario:
A user visits your website at
mysite.com. The user's machine queries their configured DNS server (most likely their home router). Their router is set to forward DNS queries to Google's public DNS server at
If Google's server has already looked up the IP of
mysite.com, it will check the time it has been cached against the TTL of the record - typically TTLs are set at 86400 seconds (1 day - 60*60*24). If the record is older than the TTL, it will discard it and attempt to look it up. The exact algorithm used to look up your records may differ (i.e. they can either use their own recursor/forward, or use "root hints"). Either way, the nameservers specified for your domain (check WHOIS if you're not sure) are where your domain's records live, and these servers will be queried if none of the servers along the query path have cached your record. Your DNS TTL is therefore the maximum time that a DNS server will cache your records, which is why DNS changes take time to propagate fully across the entire Internet.
You mention your webhost's DNS servers went down. This can lead to a multitude of problems, depending on what exactly failed.
If the DNS servers that your own servers use for name resolution went down, your web-server would not be able to resolve anything, which would mean that they would still respond to others, but any DNS lookups (most likely reverse PTRs) would fail. This would break any "phone home" functionality in Drupal or WordPress for example.
If the webhost's nameservers failed, this would mean that any new queries for your domain's DNS would fail - cached results would succeed due to the TTL settings above, so not all of your users would necessarily experience problems.
Bottom line: if you're concerned about your domain not being resolvable if your nameservers fail and you're not too worried about losing the ability to quickly change your IP, you should look at raising the TTL of your records. If in doubt though, get in touch with your webhost.