Let's start from the question and work our way into your suspicions.
How Important Is Conditioning For Weightlifters?
By weightlifter, I presume, you mean a person who works with weights for the purpose of increasing strength.
If you only care about that - conditioning, namely cardio, is not really important and doesn't affect your strength gains.
Starting Strength book talks about it and is a generally good reference for building strength for beginners (and unless you actually do strength competitions - you are a beginner).
Now, there are a lot of unknowns here. We don't know your program, your rep range, your schedule, your working weights, etc. It could be that you are doing some movement incorrectly, or working in the rep range that doesn't benefit strength gains in some particular areas. You also mentioned that you have problems lifting more weight with lats, but ok with "upper back". It's a bit confusing because, generally speaking, lets are a part of your upper back.
Nevertheless, I suspect that you care not only about strength gains, but for other things as well, like hypertrophy, health, mobility, general fitness, etc.
Cardio may be useful in several of those areas of fitness and well being. But that's another question :)
In order to fix any plateau that you may have in strength training, I strongly suggest the book in the link. It is very detailed and thorough with basic barbell movements and programming.
Best of luck!