Trouble removing a(n old) 700x23c tire I've been riding an early 90's Italian steel timetrialer (D'accordi TT0.9) for a few months, front tire has a slow leak that needs to be replaced.

The problem is that I can't get the bead totally over the rim.  

It's the front wheel, a Campi (it says Lambda Strada on it somewhere), the tire is a 700x23c DSI I have no idea how old it is, though it doesn't seem to have a massive amount of wear.  I'm a bit of a noob on the bike maintenance front (trying to correct this), so while I think I understand how to get a tire off a wheel, perhaps I'm going about it incorrectly. 

Basically I've used used a couple tire levers and gotten about 6 inches of tire over the rim, I'd post photos but I'm new so...), but cannot for the life of me get the lever to pick up the bead anywhere else on the wheel.  While I know from reading other things that other tires on these rims may well be easier to work with in the future, I'm currently faced with the more immediate problem. Does anyone have any advice for getting the rest of this tire off the wheel?

Update: Adding that, I can see the rubber tearing away around the bead a bit I try in vain to get the lever under the bead. Do know if this is relevant to my problem, but it is a bit worrying ...
How about grabbing the tyre where it's started to come off with some pliers. Then put a foot on the wheel, and pull hard.

Good luck! I know this can be really frustrating, because I had a tyre that was tightly stuck before. If it really won't come off, buy a new tyre that looks more forgiving, and chop the old one off with some wire cutters or a knife.

I decided to replace my too tight tyre even though I eventually did manage to get it off and back on because I considered that if I ever had a flat away from home and all my tools, there would be no way I could repair it, and I didn't want to risk being stranded a long way from home.