In Wizards of the Coasts article “Prestige Classes and Rune Magic” states that it uses the normal D&D multiclassing rules. Comparing it to what I know of D&D 3.5e prestige classes consist usually of four levels. Does this mean that in 5e, a character can only have 16 levels in a base class and 4 in the prestige instead of 20 levels in a base class and 4 in the prestige?
Your question boils down to two ingredients: Do prestige classes work like normal multiclassing, and does D&D 5e still have a level limit of 20?
Do prestige classes work like normal multiclassing? Yes:
Mechanically, a prestige class is a character class that requires the same training and focus as any other class. You enter a prestige class by way of the normal multiclassing rules.
Does D&D 5e still have a level limit of 20? Yes.
As a side note, the first prestige class we've seen (the Rune Scribe) offers 5 levels to multiclass into, not 4. It bears repeating that experience with mechanically unrelated games can be misleading and should generally be ignored when looking at how D&D 5e works.