Are failures cached?

[Jeff Lightner]

We had an issue last week where our Registrar displayed incorrect DNS
servers for our primary domain for several hours before we realized it
and corrected the issue.  We're still investigating how the original
problem occurred but my question isn't about that.   There was no spoof
site corrected as attempts to resolve our domain website or its MX
record returned no information as it wasn't on the (wrong) target DNS
servers.
The issue was that it appeared many people (including AT&T) seemed to
fairly quickly get the wrong information so quit sending us email but
took 2-3 days to get the right information once we'd corrected the
situation.   Are failures to resolve domains cached?  If so are they
cached longer than actual domain records?  Would the TTL for the wrong
DNS server be used for a domain for which it wasn't authoritative when
determining how long to keep a cached record?
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