This is a great question, but not one that can be answered with "yes" or "no".
In some areas, it is known that the professional grade tools and ingredients require much skill to use well and so the general public is better off using consumer grade tools. (Try tasking the average office drone to create a report with Word or QuarkXpress and you'll see what I mean). But there is no such general rule in cooking. And yet, this does not mean that better quality meat will always make better quality food. It depends on the details of each situation.
First, what does "high quality meat" mean? A filet mignon is a prized cut, and costs much more than shank. But even the best cook cannot make a good stew out of filet mignon. So in the first place, you have to know which category of meat is suited for the dish you are making. Only within a given category will better meat produce better results.
Second, "quality" means different things to different people. A cooking beginner usually cannot recognize which meat will produce better results in the given category and goes by factors like price and store reputation to decide which meat must be better quality. This is not necessarily correlated to better taste. So, as a beginner, you can pick meat you believe to be better, but end up with worse results.
The combination of these two points means that you should be aware of the difference between a "better" cut and a higher-quality version of the same cut. A "better" cut is actually a different cut, so it's a substitution that usually won't guarantee a better result. But a higher-quality upgrade of the same cut is likely to make a dish better. So, you need the cooking skill to tell a given piece of meat will be a good substitution for your earlier choice (a higher-quality version) or not (it is simply a better cut). Also, if somebody is trying to sell you a different version, you need to be able to recognize its quality (marbling in steaks, color intensity in pork chops, etc.)
Third, you can certainly find many cases where the more refined dishes, which need rarer, prized cuts, require a lot of skill to pull off well. So, if you decide to buy the "better" meat - as in the filet mignon vs. shank example - and have enough knowledge to match the dish to it, your skill might not be sufficient to pull off the dish for that meat well, while it would have been sufficient to make a good dish of the type for which the cheaper meat was suited.
Fourth, for each dish, the relative influence of skill vs. ingredient quality will be different for the final dish quality. So, it is possible that for going from "horrible" to "kinda edible" you need to get some skill, no matter what meat you use, but the jump from "kinda edible" to "not that bad" requires better meat, not better skill.
It is impossible to list all possible situations, dishes and meat cuts and say for each where the end result quality is skill-limited and where it is ingredient-limited. I am afraid that you have to be an intermediate cook before you can recognize if a mediocre cook can improve his results by cooking a given dish with better ingredients.