Will we run out of silver deposits to mine? No, I think that's a reality we likely will not face in our lifetimes.
Will we run out of silver deposits to mine at current production costs? Yes, absolutely. The cost of production will continually rise as deposits are less rich and less common. Of course, we also have to consider things like oil prices, alternative energy sources decades in the future, etc, but I think production costs will continue to rise.
Furthermore, because the price has been suppressed so low for so long, and because above ground reserves have been dumped on the market to the tune of billions of ounces over the last 50+ years (which has distorted supply and demand interactions), miners now are simply not prepared to keep pace with rising demand. Because of these low prices, they simply are not investing enough money into exploration and the creation of new mines.