I use an R9 285 currently, which is rated at 190W vs the RX580's 185W. I have a 460 Watt power supply, providing 385W on the 12v rail. My CPU is an i5 Haswell rated at 84W.
I don't know what your current CPU is, so I don't know how much power that draws. Nor do I know how many DIMMs you have, how many fans, whether it's a big or small motherboard, etc.
I would probably think hard before considering putting a Vega 56 in with the existing power supply... I've been doing a few calculations, and I think adding a pair RAM sticks might theoretically push me over the edge in that regard - though, I know my motherboard power estimate is VERY much guesswork, and I probably could discount the power draw of the optical drive, because it has seen use maybe once or twice in the three years I've had this PC.
If the Ryzen 5 1600 is a significant drop from your current CPU's power usage, then, possibly. Still, I'd give yourself a little more cushion and go with a higher rated (and high quality) power supply.