You can certainly go higher and I wouldn't recommend the 16 series anymore
The lack of hardware RT support and also being limited to FSR mostly isn't worthwhile the money you pay for those cards

Not to mention that you can put up even a 5090 in your system if you want (provided your case and PSU can handle it) but you will end up with reduced performance as the CPU will hold back that GPU
But overall, the CPU will very comfortably handle an RTX 3060 12GB or the RTX 4060Ti 16GB for that matter just fine. of course, those are just examples for the question

I wouldn't recommend an 8GB card for simple reason of modern games do require 8GB of VRAM as the minimum and what happens when you are out of VRAM? you go to system memory through the PCIe interface. since your platform is limited to PCIe 3.0. you will have a much larger performance drop compared to a modern system with PCIe 4.0/5.0

Originally posted by nexx:

One recommendation I got was the NVIDIA GTX 1650 or 1660 but could I got higher than that? I'm willing to get a RAM stick/upgrade as well if that needs changing

It's all about price. I personally would recommend an older RTX (2070 for example) card to get access to DLSS4 transformer model. You don't even need to use upscaling - just use DLSS in native mode (DLAA) for AI based Anti aliasing instead of blurry TAA.
Then use Optiscaler again or Lossless Scaling for frame generation if you like. It's a very flawed solution but it's decent at mitigating CPU bottlenecks. With Lossless Scaling you can even have multi frame generation. 6x 10x or even 20x. Better than the 5090 can officially do. ;)

It depends on the game and what fps you are hoping for.

I have a 6700k and 3060ti and the cpu does struggle with some games, but not all. I thought about a 4070 for the extra vram but I’m now glad I didn’t spend the extra money.

The suggestion above of a 3060/4060ti is a good shout if you want the vram but you’ll definitely be holding back a 4060ti. The 3060 is a huge upgrade for you but a reasonable step down compared to 3060ti/4060ti. For some games, you might wish you bought something a little more powerful.

well the 1060 or 970 would work.. but if you want something a little newer in generation but with similair performance..

the radeon 6500 xt would be a suiting fit. 8GB models do excist.. which I than would pick for 4gb vram really is not enough anymore.

don't expect stellar performance.. but certainly much better than the junk of on cpu grafics and 1030

now I do have to add just because something HAS a bottleneck does not mean it is a ceiling.
that 3060 is nearly twice as fast as a 6500xt.. and sure while it is bottlenecked by about 25%
75% of 200% still is 150%..

so you might go a bit beyond to something like a 6650xt that generally has a favorable pricepoint

do note though that running bottlenecked with an 6650 or 3060 will lead to higher average fps.. it will also increase worse 1% lows and tearing.. vs a non bottlenecked card like that 6500xt..

now as for prices..

I see a new 6500xt 8gb for 170 euro.
I see a new 6650xt 8ghb for 220 euro

I see complete 2d hand systems with a 6650 or 3060 in them and a much newer cpu than you got around 250 euro...

so yeah it might be time to just get a whole 2d hand system.. instead of just sticking a better card in it... by the point you are spending that kind of money..

psu brand/model/age?

would be the limiting factor

the cpu would be limiting max fps
but if the res is 1440p or higher, the gpu would more likely be the limit

lower res and visual settings, to see max fps cap by the cpu or game engine

Is this a pre built pc?

seen 3060's for a 100 bucks used.....that is about what i would do

depending on what games and resolution you play

your 10 yr old 4core/8thread cpu would probably be good for up to RTX 2080 maybe rtx 3080

seriously its not even overclockable