This bug is related to these other bugs:  3959, 6160.

Alexey Kuznetsov has recently posted a new copy of his
iputils package from which RedHat's 'ping' is taken.
Located at ftp://ftp.inr.ac.ru/ip-routing/, the package is
named iputils-ss991015.tar.gz.  This update fixes a bug with
the new -w # (timeout) option which exists in the Red Hat
6.1 release but doesn't work as advertised in the man page.

A little history:  Bug 3959 filed against Red Hat 6.0
reported that "ping -c #" no longer exits when pinging a
down host, and this was marked as resolved in the next
release.  Alexey apparently introduced the -w option to
compensate for this same problem, but in Red Hat 6.1, the -w
command does not work as intended due to a bug in the code.
As a result, ping still never exits when talking to a down
host.  Bug 6160, also filed against Red Hat 6.0, appears
to be a duplicate of 3959.

There are two possible solutions to this bug in Red Hat 6.1:
One is to install the new ping with the working -w option;
the other is to look into fixing -c properly.

I find it curious that -w was added rather than fixing -c
to exit when a host is not responding, thereby keeping
things compatible with existing software that expects ping
to behave in a certain way, but I suppose this matter would
have to be taken up with Mr. Kuznetsov.