I know. You've heard this question asked and answered a million or so times. The standard answer from professionals and amateurs alike is a flat and simple, “No.” In my opinion, though, there's more to both the question and the answer. In fact, I think that answer is frustrating for a lot of amateurs and those considering photography as a hobby or profession and I'm going to explain why.
There's a phrase you've no doubt heard referred to as one of the things that photographers hate to hear: “Wow, what a great shot! You must have an awesome camera!” Granted, it may be insulting for someone to assume that your talents had nothing to do with it, but in some cases, it may be that the gear really should get a little of the credit, too. I mean, it's easy for someone shooting with, say, a Canon 5DS or Nikon D810 to say, “The camera didn't create this image,” but maybe it's worth considering whether the final image would have been possible without that camera.
Here's a more specific example: Let's say you've captured an incredible bird photo with your Canon EF 500mm f/4L IS II. You're the one that took that photo. The lens had nothing to do with it, right? Well, no. The truth is, had you tried that shot with a 75-300mm kit lens, the results would probably have been very different. Not only would you need to get closer, increase your ISO setting (increasing the noise level) and perhaps your aperture size as well (shortening the depth of field), but you probably would have had to deal with CA and diffraction issues, too.
I could go on citing examples, but you should be getting the picture (pun intended). Now, put yourself in the shoes of a novice, with an entry-level rig, listening to a pro saying, “You don't need the best gear to be the best photographer.” That's easy for him/her to say, while sitting beside a couple of full-frame bodies and an arsenal of great lenses. To be fair, that's probably an unjust stereotype and I apologize to any pros I've offended, but the point is, there are limitations that can be attributed to the gear that you use.
There are reasons for categories like “entry-level” and “professional-grade”. All it takes is a chance to try both to understand those reasons. Shooting with pro-level equipment won't make you more creative. It won't give you a better eye for composition or further your knowledge of exposure settings. It won't correct your mistakes. What it will do is help you showcase your skills and talents with superior results from your efforts. High-end gear makes more shots possible and allows you more flexibility in processing and printing.
So, in my opinion, while the answer to the title question may, in fact, be “No,” it needs further qualification. Better gear may not make you a better photographer, but it can help you get better results with your creative efforts and technical skills. Perhaps the best piece of advice is another one you've heard before: "Buy the best that you can afford." Keep in mind that one way to upgrade is to buy good used equipment. Whether you choose new or used, I recommend buying from one of the most respected dealers in the USA: Hunt's Photo and Video.