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Currently I have the following setup:
Ryzen 5600 with TT 310 cooler
AMD RX 6600
16GB 3200 Mhz DDR4 (8*2)
500 GB Nvme SSD+ 1TB SATA SSD
PSU: Thermaltake SMART BM2 550W 80 Plus Bronze Semi-Modular
22 Inch IPS 1080p Monitor 75hz
Can I run a 5060ti 16GB without upgrading my PSU?
For power (PSU) requirements you need to consider the entire build.
FYI:
https://www.tomshardware.com/reviews/best-psus,4229.html
Do some estimates by using the calculators at the end of the review.
And then do your own manual total based on component power specs.
If any given component provides a wattage range then use the high end wattage value.
For the most part, I would expect 550 Watts to be too low.
You need a quality PSU (costs more) and be sure to use only the cables that come with any given PSU.
Back up all data before doing or attempting to do any upgrades.
Thermaltake SMART BM2 550W 80 Plus Bronze Semi-Modular
That'll be a firm no, the Thermaltake Smarts are very low quality units.