Spain headed into Euro 2016 having won the previous two European crowns either side of triumphing at the 2010 World Cup.
But Spain suffered a last-gasp loss against Croatia to slip to second in Group D and set up a last-16 clash with Italy, who gained revenge for their humbling 4-0 defeat in the final of Euro 2012 thanks to goals from Giorgio Chiellini and Graziano Pelle.
"We wanted to keep going, but we ended up with the most difficult route and we weren't up to the level required," said Barcelona defender Pique.
"There are many circumstances, the level of the players, a mixture of things.
"After the defeat against Georgia I said we weren't favourites and I think that right now we are not among the best. We have great players but we are not at the level to win tournaments."
Pique wants this bitter experience to motivate Spain for the challenges ahead, while he would be happy for Vicente del Bosque to stay on.
The veteran coach, who oversaw Spain's marches to glory in 2010 and 2012, was widely expected to retire after the tournament but told a post-match news conference that he would take the time to discuss his position with the Royal Spanish Football Federation (RFEF).
"He has earned the right to carry on. It's his decision," Pique added.
"This has to help us to prepare for the 2018 World Cup in Russia and know our place, that we are not favourites and that we have a lot of work ahead to prepare for Russia.
"It's not the best moment of our careers. Now we will rest and reflect and we'll see what we'll do next year."