I have an entry in my crontab that looks like this:
0 3 * * * pg_dump mydb | gzip > ~/backup/db/$(date +%Y-%m-%d).psql.gz
You need to escape "%" characters in crontab entries with backslashes- see the crontab(5) manpage. I've had exactly the same problem.
0 7 * * * mysqldump usblog | bzip2 -c > usblog.$(date --utc +\%Y-\%m-\%dT\%H-\%M-\%SZ).sql.bz2
Do you not get emails of cron errors? Not even if you put "MAILTOfirstname.lastname@example.org" in the crontab?
You may also need to set PATH in your crontab if pg_dump or gzip isn't on the system default path (so use "type pg_dump" to check where they are, crontab usually only runs commands in /bin or /usr/bin by default)