Hamilton said: "I was expecting us to get at least fifth, but those teams in front of us were much quicker.
"We will keep our heads down, stay focused and hopefully at some stage we will get a car to challenge at the front."
Button added: "You can see where our weaknesses are. I just don't think we are very quick at the moment. I don't think we did anything wrong with strategy, we just were not quick enough today.
"We need to find some pace. It is not just the Red Bulls and Ferraris who are quicker than us, a lot of cars are. We have a lot to work on."
McLaren's Silverstone performance marks the confirmation of a dramatic turnaround in form for a team that started the season with the fastest car.
Hamilton took his only win of the season so far in Canada a month ago, but the car has been uncompetitive in the last two races at the slow Valencia track and now high-speed Silverstone.
Season so far: in numbers
- McLaren have scored the most podiums of any team, from a maximum of 16, one more than Lotus and three more than Red Bull.
Team boss Martin Whitmarsh admitted McLaren's lack of competitiveness was a concern.
"I'm not seriously worried," he added. "Worry doesn't make your car go quicker, does it?
"[I'm] Disappointed, but there is no magic, you have to work on developing the car and understanding the conditions."
Whitmarsh said a better result would have been possible for Hamilton but that the 'hard' tyres they fitted for the final stint did not perform in the same effective manner as a set of the same tyres had in his more competitive first stint.
"What you can do is learn about the tyres, do a better job and bring the upgrade packages as quick as you can," Whitmarsh said.
"We have a reasonable upgrade package for Germany and we'd like that to be as successful as we can and make sure we can make it for Germany.
"It's a tough championship, but we have seen lots of swings in this championship so far.
"It's an exciting championship, [the] right thing at the moment is work hard and make sure operationally we do a good job - make sure we make the car go quicker. Gain understanding of the tyres and get on with it."
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