Possible but quite unlikely. If Bitcoin were to replace all the fiat in the world in every single country, it would reach about $1 million per coin for a total market cap of about $21 trillion (actually it's slightly less than this due to some coins being lost over time). In that case, 1 satoshi would still only be worth 1 cent.
However, if this happens in the very distant future then it's possible that USD could lose 99 percent of its value due to simple inflation. If this happened, 1 USD by then would be worth 0.01 USD today. The dollar has already lost 96 percent of its value since 1913 so it's not impossible. So perhaps once Bitcoin is fully mined in the year 2140, 1 satoshi being worth 1 USD might be possible. Not that it really matters however since a coffee would cost you $300.
And finally, there is also the risk of hyperinflation. It happened in Germany. It happened in Hungary. It happened in Yugoslavia. And most recently, it happened in Zimbabwe. It could happen in the US too. This would have the same result as above although it would obviously happen much faster.
Additional note: If Bitcoin reached a status comparable to that of gold, then it would reach roughly $350,000. That would be a more realistic scenario.
What's important is to try to calculate what will be the purchasing power of 1 BTC in the next 10 or 20 years, not the parity with dollar which could indeed decrease a lot. The question is: how??