It's all in the pixel dimensions of the image. If you have enough pixels then it doesn't matter what the resolution is. Generally speaking, 300 PPI is a suitable printing resolution. So simple math can tell you how many pixels you need. In your instance, you are looking for a 16 x 20" print. The math would be as follows:
16 inches X 300 PPI = 4800 pixels
20 inches X 300 PPI = 6000 pixels
So to print a 16 x 20" print ad 300 PPI you would need an image that is 4800 x 6000 pixels. It doesn't matter what you have set in the resolution because the resolution setting really has no impact on quality. It is simply there to help programs such as Photoshop to calculate how large the image will be at a given resolution.
In reality, you probably don't need 300 PPI for a 16 x 20" print. The viewing distance will be greater, and you can probably get by with 240 PPI or even 200 PPI. So, do the math as in the example above, and you would be able to determine how many pixels you would need to print suitably.