As always, you guys are the best. You are right and the
fault is on our end.
For some reason, it seems that the 3 way bulbs got a bit loosened. They
were turning on and off, but not three way.
We cut off the inline switch first, and connected the cut
wire. The lamp was still one way. I went to look at lamp socket,to make
sure everything was tight, and gave the bulb about a half a turn. That did
the trick. It seems if the bulb is not completely tightened it would turn on
and off, but lost three way.
So we had to put the inline switches back on again and
everything is fine.
Never had 3 ways that were that sensitive, that they would work
one way, unless sufficienty tightened. It must have something to do with the
new sockets we put on, and they really need a very tight bulb fit to work 3
way.

Thanks for everyone who helped.

That simply is not physically possible.

As the secondary poster wrote:

"Are you sure that yoiu are using three way bulbs?"

and

"Are you sure you installed a three way socket?"

Post and provide a link to pictures of the bulb base and the interior of
the
replacement socket you installed.