Why can my EC2 instance no longer connect to itself using its public DNS entry?

Why can my EC2 instance no longer connect to itself using its public DNS
entry?

One of our Amazon EC2 instances can no longer connect to itself using its
public DNS . It says "unknown host". Previously, when used from within
EC2, the public DNS resolved to the current internal IP. This is still the
case for another instance we also use.
This has worked for the previous 8 months. Last week, our continuous
integration server on that machine ran into the problem a few times. After
the instance was started today, connecting via public DNS no longer works
at all, even though the elastic IP is definitively attached to the
instance.
Pinging that public DNS entry from elsewhere - both from inside and
outside EC2 - works perfectly fine. The instance can ping itself using the
elastic IP, but we want to connect directly. The instance can also ping
itself using its current internal IP or internal DNS entry, but as that
changes on startup, it is not a solution.
As we did not change the machine configuration in any way for the past two
months, I am reasonably sure this is a problem on Amazon's end. Can anyone
confirm this, has experienced similar issues or a suggestion what to do to
fix this?
Note: we only have the "basic" AWS suppport level, so I can't file an
issue with Amazon about this. I did post this question on the AWS forums,
though.