In addition to Michael Clark's answer that answers whether it is the equipment that makes good photographs or the photographer, I want to answer the question How to know you've outgrown your equipment?
Beware: I will offer my life story as an anecdote to answer the question. It will be long and probably (yet hopefully not) boring. For all of you who want to skip that: I wrote a conclusion that precedes my lengthy life story. There is another conclusion at the bottom, which is a bit broader. Or, even easier: Read Crazy Dino's answer.
Conclusion (a.k.a. "TL,DR")
Today's equipment is hard to outgrow of: Even the most basic DSLR of today offers better low light- and AF-performance than any professional SLR Exaggerated, not empirically proven statement!.
Of course, there are ways to outgrow your equipment: The easiest way is to become a professional whose financial situation is entirely dependent on getting every shot right.
With lenses, things are easier: Focal length will limit your framing (try to get a close-up shot of a tennis player in-game with a 8mm fisheye - or try to get a whole concert hall with a 400mm tele prime lens). Aperture will not be as limiting as focal length, but still, in low light environments, a wider aperture might be obligatory. (Fast) autofocus and image stabilisation are nice-to-have, too, as is sharpness, though these things all can usually be compensated for.
(Anecdote) I think that I've outgrown my equipment
After messing around with analog SLRs for a while, I started with an EOS 450D and its EF-S 18-55mm kit lens plus an EF 70-210 f/4 that I borrowed from my father. I was quite happy with this setup, and although sometimes, there was a shot that I did not get, overall, I was quite impressed with it.
After 2 years and 360 days, I bought an EOS 60D after about half a year of decision making / Gear Acquisition Syndrome. I was impressed by the 60D's faster fps, its cross-type AF sensors and the additional (compared to the 450D) quick control dial. Naturally, I immediately thought that the lack of those features had been the reason that I could not take better photographs. As money was scarce, I bought myself a Tamron SP 17-50mm 2.8 XR Di II VC as my first/"always-on" lens.
That was the unluckiest decision of my photographic carreer (so far ;-) ). I quickly found out that I did not need more fps, that cross-type AF sensors are not some kind of magic bullet, and that overall, there was not a single thing that I saw as improved over the 450D (okay, the video recording stuff was nice to have).
But here's the thing: It would be easy to say "bah, the 60D was a bad camera, anyway". Although I despise1 it, I am quite sure that it is not. I simply bought the wrong tool for what I needed and for my level of skill. I thought that by simply upgrading the hardware, I could get better shots - the very thing that Michael tells us to be unrealistic in his answer.
(Anecdote) I learn that I've not outgrown my equipment
I did say that I "quickly found out" that the 60D did not deliver any ground breaking changes to me. However, the good thing about that was that I now had a camera that I did not like1, but that I bought with my own money, so I had to stick with it for at least a few years. During those years, I learned to overcome the weaknesses1 of the 60D. I improved both my overall style and my technical knowledge to a level where I really found practical limitations to the 60D - and ways to overcome them (where humanly possible).
In that era, I bought a Tamron SP 70-300mm 4.0-5.6 Di VC USD that I still own, and an EF 50mm f/1.4 USM that I sold without hesitation after just one year (it seems that 50mm on APS-C is not my favourite focal length).
Almost 3 1/2 years later, I bought an EOS 5D Mark III (which I still own) with its EF 24-105 f/4L USM kit lens. Since then, I had to work (professionally) with many 3-digit and 2-digit DSLRs plus an EOS M6, and I cannot remember what it was that made me believe that I could outperform an 450D. Granted, full frame offers less noise and ultra wide angle lenses are (were?) easier to get, but apart from that, I only find that some comfort features are missing in the "lower class" DSLRs. Seems like I learned to adapt to my equipment faster - and to accept its limitations.
I later bought a Samyang 14mm f/2.8 and an EF 100mm f/2.8L IS USM Macro - both of them are among my favourite lenses. Today, I find that the most limiting factor to me is the lack of more lenses. And while I am thinking of upgrading to the next high-resolution full frame EOS, I will certainly wait for the next generation or even two: As I learned to both accept and compensate my equipment's (very minor) quirks, there is no need for me to go full Fry about it.
Conclusion (in depth)
I think there are no tell-tale signs whether you have outgrown your equipment or not. In all honesty, I do not think that one can outgrow their equipment nowadays - even entry level DSLRs and MILCs from today are far superior to most analog stuff. The only way to outgrow it is if you need a feature that your current equipment does not offer: If you are a sports photographer, then 6 fps probably will not be good enough (well, it is, but double the fps and you double the chance for a perfectly timed photo in your burst) and you will want the best AF system you can afford. If you are into landscapes, then perhaps you want the highest resolution you can afford (though a good tele lens and a good tripod can do the same with any camera - if you have time for stitching). If you are a travel photographer, then a light, universal setup might be the best choice. Etc. p.p..
Another way to outgrow your equipment is to become a (serious) professional: If one single missed opportunity is enough to get you sacked forever, then you will want to buy the most resilient equipment there is.
Because of all of that, it is very likely that at some point, you will buy equipment that does not help you improve your shots. However, there is no certain way to tell (as a third party). If you now decided to buy a 1D X Mk II, I would think that to be overkill, yet you may have your reasons for the upgrade (and I might just be jealous ;-) ).
Now that is far easier, as anyone can see the difference between an EF 50mm f/1.8 and an EF 100-400mm f/4-5.6L IS USM II. No, I am not talking about the color (including the red ring): I am talking about the focal length and a bit about the widest available aperture.
If you want to do portraits, then both an 8mm fisheye and a 600mm tele prime probably are not the right tools. If you are into birding, then you might want the longest focal length you can afford. If you are mainly photographing in low light environments, then you will want to buy the fastest lens you can afford. Etc. p.p..
Still, you might buy the wrong lens some time - e.g. I bought the EF 50mm f/1.4 USM just to find out that I do not like it at all.
1 Please note that I still hate the/my 60D. This is a 100% subjective feeling that is not backed by any objective measurement and in reality is some kind of projected self-hate for buying the wrong camera with what little money I had back then.