If you really don't want the second command to proceed until the first is known to be successful, then you probably need to use temporary files. The simple version of that is:
tmp=${TMPDIR:-/tmp}/mine.$$
if ./a > $tmp.1
then
if ./b <$tmp.1 >$tmp.2
then
if ./c <$tmp.2
then : OK
else echo "./c failed" 1>&2
fi
else echo "./b failed" 1>&2
fi
else echo "./a failed" 1>&2
fi
rm -f $tmp.[12]
The '1>&2' redirection can also be abbreviated '>&2'; however, an old version of the MKS shell mishandled the error redirection without the preceding '1' so I've used that unambiguous notation for reliability for ages.
This leaks files if you interrupt something. Bomb-proof (more or less) shell programming uses:
tmp=${TMPDIR:-/tmp}/mine.$$
trap 'rm -f $tmp.[12]; exit 1' 0 1 2 3 13 15
...if statement as before...
rm -f $tmp.[12]
trap 0 1 2 3 13 15
The first trap line says 'run the commands 'rm -f $tmp.[12]; exit 1
' when any of the signals 1 SIGHUP, 2 SIGINT, 3 SIGQUIT, 13 SIGPIPE, or 15 SIGTERM occur, or 0 (when the shell exits for any reason).
If you're writing a shell script, the final trap only needs to remove the trap on 0, which is the shell exit trap (you can leave the other signals in place since the process is about to terminate anyway).
In the original pipeline, it is feasible for 'c' to be reading data from 'b' before 'a' has finished - this is usually desirable (it gives multiple cores work to do, for example). If 'b' is a 'sort' phase, then this won't apply - 'b' has to see all its input before it can generate any of its output.
If you want to detect which command(s) fail, you can use:
(./a || echo "./a exited with $?" 1>&2) |
(./b || echo "./b exited with $?" 1>&2) |
(./c || echo "./c exited with $?" 1>&2)
This is simple and symmetric - it is trivial to extend to a 4-part or N-part pipeline.
Simple experimentation with 'set -e' didn't help.