It is definitely possible to improve more than 10-15 points, so don’t give up hope when you’re just getting started! Here are two pieces of advice:
(1) One of the most important parts of studying for LSAT is carefully reviewing an exam after you take it so that you can learn from your mistakes. For each question that you got wrong, make sure you understand why the answer you chose was incorrect, and why the correct answer is correct. There are really good (and free) explanations for a lot of the exams over at http://lsathacks.com/ You can also post questions here at lsatters if you don’t understand why you got something wrong.
(2) When you take a practice test, or just a timed section, make sure not to rush through it. You do not need to attempt all of the questions in order to get a good score- give yourself enough time to answer the questions that you do attempt correctly. For example, your first goal for the logic games section should be to be able to complete one logic game, with perfect accuracy, in 35 minutes. Your first goal for reading comprehension should be to be able to complete one passage, with perfect accuracy, in 35 minutes. Your first goal for logical reasoning should be to be able to complete, say, the first ten questions (or maybe less, depending on where you’re at right now) with perfect/near-perfect accuracy, in 35 minutes. I’m guessing that this is probably one of the things that went wrong for you on the diagnostic- you were probably trying to answer as many questions as you could, instead of taking your time and focusing more on accuracy.
Good luck, and feel free to ask us lots of questions! 🙂