We sailed on the Sapphire in October. When we first boarded we made reservations for eight (two tables for four each) at 5:30 for the entire cruise at Sterling Steakhouse. When we arrived at 5:30 the restaurant was practically empty. When we were leaving at about 6:30, it was just starting to get crowded, but there were still tables available.
You don't mention which ship you will be on. The Sapphire (as well as the Diamond) are a bit different than the other Princess ships in that they have taken the two personal choice restaurants available on the other ships and divided them into four smaller restaurants each serving a themed specialty menu. I believe Sterlings is the most popular of the four, and from what I seen, I don't think you would have had any trouble getting a table there before about 6:30.