If you search the Forum Page for the keyword wallpaper you'll find LOTS of 'hits' that talk about different 'issues' with doing this job.
The one very recent post that talks more directly to the 'troubles' you're having is at Wallpaper Removal. Do check it out. Maybe you're not perforating the paper/vinyl enough with the Paper Tiger. Vinyl, especially, is difficult because unlike paper-wallpaper, it doesn't absorbe any of the DIF mixture. Hence, you'll want to make a LOT of perforations with the Tiger so the solution gets back there.
I wouldn't use a Putty Knife AT ALL to do this job. Use a 4" Spackle Knife. Let the DIF mixture of BOILING, HOT water (be sure to wear rubber gloves) 'sit' on the perforated paper for the instructed 15 minutes. You'll need to do it twice you know. Then gently scrape. BEFORE you put on the 2nd 'coating' of the DIF mix, go to a 2nd wall that you need to strip and put on its FIRST coat. This way, you're 15 minutes 'ahead' of yourself (if you know what I mean.) Then, right away, coat the 1st wall with its 2nd coating. BEFORE you scrape the 1st wall, put your 2nd coating on your 2nd wall. ANd so on. If you pace yourself well AND let the DIF do its job, you'll be fine. There's absolutely NO problem with putting a 3rd or 4th coating on the wall to loosen the paper up. Also, I found it very beneficial to put a 3rd coating on the wall AFTER the 2nd coating had its 15 minutes to 'do its thing'. The 3rd coating is applied and I IMMEDIATELY start removing the paper. You see, during the 'working time' of the 2nd coating, it tends to start drying up. Soooo, by wetting it again JUST before you remove it helps to 'loosen' it for the short time you'll be stripping off the paper.
Good luck! If anything I said isn't clear, e-mail me directly or post back up. My best to ya and hope this helps.
Jay J -Moderator
PS: DO keep a pot of HOT water on the stove. Keep the flame low so you don't burn the house down! (But it should be BOILING hot before you use it.) DON'T be afraid to make a 'fresh' mixture! After a while, the mixture in the bucket get's 'polluted' from the constant use of the sponge being dipped into it. THis, too, is another common reason for the DIF mixture not working for folks. The DIF is cheap enough. Buy MORE than you think you need - You can always return UNUSED containers.