There is no limit on the colors you can add to a postcard. However, that depends on the ink you’re using. If the printer uses CMYK colors, there is no limit; but there is a limit if the printer uses Pantone colors.
In printing, there are two different ways to print: digital printing and offset printing. Digital printing uses CMYK colors, and since it is a digital printing technique, it can print as many colors as needed, although colors tend to vary in each set (not much, but they vary).
On the other side, we have offset printing. Here, printers need to prepare the inks manually (each Pantone color). This process creates limitations on the number of Pantone colors you can print on a single postcard, varying from one to five. Five being the maximum colors you could add to a postcard without making it unaffordable.
To give you an example of when to opt to use offset printing and not digital printing, we’ll use Coca-Cola as an example. Coca-Cola’s branding is exceptional, and one of the reasons it is like that is because they always use the same red color, no matter where you find it. To make sure their red color always looks the same, they use a Pantone color and print it in an offset printing machine; that way, there are little to no variations in how the color looks. If they used CMYK colors, variations would be more common.