So I have been tuning up an old 1986 Fuji with Suntour Lepree components. After a bit of trouble the front derailleur is a-okay but I have had immense trouble getting the rear derailleur going. I set the H-limit screw with no problems but found I couldn't shift up past the 3rd smallest gear (on a 6 gear rear wheel). The L screw wasn't touching the stop when the lever was pulled all the way back, so I'm wondering how I can get that sorted out, I'm thinking adjusting the cables might work but it's not clear to me what's causing it.

Any help would be appreciated.

Sounds like a cable adjustment issue indeed. I would start from stratch.

It sounds like you may have left the shifter in the 3 or 4 position when you tightened the cable anchor bolt. Also I'd make sure all the housings are inserted in the frame cable stops properly.

Let us know how it works out.

+1 If I read your post correctly, you can't shift onto the largest cogs (that is, into your lowest gears). If your limit screw isn't the problem, then it's likely that your cable doesn't have enough tension. Tighten the cable by the proper amount, and you should be fine.

That's what I figured, I just bought the bike used so I haven't yet touched the cables yet, I was tuning it figuring all I needed was an adjustment of the limit screws.

Update: After some fine tuning the derailleur is humming along. It makes the tolerances on the front derailler pretty tight but it's all smooth as silk now.