PARIS—It may be too early to sound the alarms in Dubrovnik and Bratislava, Fulnek or Wuhan, but “Here come the Americans!” will have a nice ring to all those fans in the United States, stressing out over where the next Venus and Serena Williams will come from.

Of course, there will be no “next” Venus or Serena, any more than there will be a “next” year 2000. Still, five women, led by Serena herself, are in the third round of the French Open—the most since six made it in 2006. They others are: Varvara Lepchenko, Jamie Hampton, Bethanie-Mattek Sands, and Sloane Stephens.

The first three of those last four women are a pleasant surprise, while Stephens’ performance demands use of the word “relief.” An overnight sensation at the Australian Open, where she upset Serena to reach the semifinals, Stephens struggled mightily in the ensuing months. She made some progress in recent weeks, winning two matches in Rome and two in Brussels. A flame-out here, though, would have been construed as a painful setback. After all, she first caught the eye of people in a big way when she reached the fourth round at Roland Garros last year. She was No. 70 then; now she’s No. 17.

Today, Stephens beat her countrywoman Vania King in an impressive display of guile and power, 6-1, 6-3. She said afterward, “I played well. I’ve been playing much better the last couple weeks. I played pretty solid, so that was good. Even the rain and the delays and everything I stayed pretty solid, so that was good.”

It was a rough day for players here, with (ultimately) four rain delays (five if you count the one that ended the day at 8:58 pm) creating havoc with the scheduling—and great boredom and stress in the locker room and player lounge. There’s no more insidious factor at an outdoor tournament than rain; it turns the players into disaffected, tense, and edgy caged beasts. And there’s little antidote for it, according to Stephens.

“Well, I mean like today, the first time I was waiting I was just like playing on my phone,” she revealed. “The second time, I went to sleep. Then, I don’t know. I think playing the game and the sleeping was a good combination. There’s just so many different things you can do. You literally have nothing to do. Should I eat? Do cartwheels? You have no idea what to do so it’s just kind of like you’re lost.”

The bad weather may be over now, with clear days to come. In their next matches, Stephens will play Marina Erakovic, Lepchenko meets No. 8 seed Angelique Kerber, Serena Williams will take on Sorana Cirstea, and Hampton draws the black marble: No. 7 seed Petra Kvitova.