Cat6 and Cat5/e have the same jack. The difference is in twist rate and distance. Are you sure you're not trying to plug it into the modem jack..? If you're not an absolute moron, then maybe whoever sold you the cables accidentally put the wrong connectors on. Most places now either carry Cat5e or Cat6 since those enable faster speeds - with 5e being more compatible with more devices.
That sounds weird about the connectors being too wide. ???
did the tool press the pins and jacket parts properly ?
I've seen some tools which did not align properly