No, because Berry Master to the rescue!
He lives on Route 123, and gives 2 berries per day. The berries he gives you are two random ones somewhere from Berry 21 - Berry 30. His wife will also give you a berry if you tell her one of the following phrases:
GREAT BATTLE
CHALLENGE CONTEST
OVERWHELMING LATIAS
COOL LATIOS
SUPER HUSTLE
The girl in the flower shop, the old man in Lilycove, and Kiri in Sootopolis city also give out daily berries as well.
However, if your internal clock stops working, they will no longer give berries, so as long as your cartridge battery is fine, you won't run out of berries.
To answer the other part of your question, there are still some berries you can permanently run out of. All of the E-Reader berries can be run out of if you lose your E-reader. The Lansat berry and the Starf berry are only obtainable in Emerald, so if you trade one over, then use it you can't get another without trading it in. The berries the Berry Master's Wife gives are one time only, and are only repeatably obtainable in Fire Red, Leaf Green, or Emerald.
Source: Bulbapedia, my experience with Hoenn.
Hope I helped. Let me know if I missed any.