Russia, the champions in 2004, 2005, 2007 and 2008, suffered an embarrassing 3-1 defeat to the Netherlands in Moscow.
Italy, who were title winners in 2006, 2009, 2010 and 2013, were also dumped out in the first round, going down 4-1 to France in Marseille.
But Switzerland, who have never won the Fed Cup, made the semi-finals for the first time in 18 years with a thrilling 3-2 win over Germany in Leipzig thanks to Belinda Bencic and Martina Hingis clinching the decisive doubles.
While the Netherlands and 2003 champions France meet in one semi-final in April, the Swiss will tackle the Czechs, who edged out Romania 3-2, in the other.
The Dutch, without a single player in the top 100, made the semi-finals for the first time since 1997 after Kiki Bertens defeated two-time Grand Slam winner Svetlana Kuznetsova 6-1, 6-4.
Kuznetsova, the world 17, had lost a four-hour marathon against Richel Hogenkamp on Saturday in the longest singles match ever played at a Fed Cup.
The 24-year-old Bertens, ranked at 106, took her Fed Cup singles record to 13 wins and one loss after the Dutch, 2-0 up overnight, claimed a famous win against a Russian team who had Maria Sharapova warming the bench all weekend.
Russia captain Anastasia Myskina said she wasn't tempted to risk Sharapova, who won both singles rubbers in the country's 2015 final loss to the Czech Republic.
"No, she had a problem with her left arm and that wasn't good. We decided on Thursday that she wouldn't play," said Myskina.
In Leipzig, 18-year-old Bencic won her third point of the weekend when she teamed up with 35-year-old Hingis to beat Germany's Andrea Petkovic and Anna-Lena Groenefeld 6-3, 6-2.
Standing 1-1 overnight, Bencic downed Australian Open champion Angelique Kerber 7-6 (7/4), 6-3 before Annika Beck celebrated her maiden singles appearance by seeing off Timea Bacsinszky 7-5, 6-4 to make the tie 2-2.
"I was cheering them on for the first four matches and then I had to go out there," said Hingis, who returned to the Fed Cup last year for the first time since 1998 in an attempt to play in the Rio Olympics.
Bencic praised Hingis who captured the Australian Open women's title with Sania Mirza in Melbourne last weekend.
"She did all the great volleys and all the doubles stuff. I was just happy I could play with her," said Bencic, the world number 11 in singles.
The Czech Republic, recent champions in 2011, 2012, 2014 and 2015, were taken to a deciding doubles in Cluj by Romania before Karolina Pliskova and Barbora Strycova saw off Andreea Mitu and Raluca Olaru 6-2, 6-3.
Simona Halep put Romania 2-1 up earlier Sunday after battling past two-time Wimbledon champion Petra Kvitova 6-4, 3-6, 6-3 before the impressive Pliskova levelled the tie with a 6-4, 4-6, 6-3 win over Monica Niculescu.
In Marseille, Caroline Garcia defeated Camila Giorgi 6-3, 6-4 to put France into the semi-finals for the second year running.
Earlier, Kristina Mladenovic saw off Sara Errani 7-6 (7/4), 6-1 to put the French 2-1 ahead after the two nations had split the opening singles on Saturday.
In World Group II, where the winners face the Group I losers in April for places in the elite section in 2017, Venus Williams guided the United States to an unassailable 3-0 lead over Poland in Hawaii.
Williams, who played in her country's title-winning team in 1999, defeated Magda Linette 6-1, 6-2 for the crucial point.
Australia beat Slovakia 3-2 in Bratislava while Spain swept Serbia 4-0 in Kraljevo.
In Quebec City, Canada and Belarus were level at 2-2.