The Premier League champions are only one point above the relegation zone after Monday's defeat at Leicester, their ninth loss of the season.
After the game, boss Jose Mourinho said his "work was betrayed" by his players.
"If you are a big player and paid like a big player, you must play like a big player and behave like a big player," said Blues midfielder Fabregas.
"I am not saying you can't have a bad season and bad games but the attitude must be spot on.
"We must always be at the top of our games and the behaviour has to be better than what we are seeing right now from every single Chelsea player."
Chelsea are 16th with only 15 points following the defeat by table-topping Leicester.
In Premier League history, when teams have had 15 points from 16 games, they have finished, on average, in 17th position - one place above the drop zone.
Mourinho says he does not believe the Stamford Bridge club are in a relegation battle.
However, former Blues winger Pat Nevin told BBC Radio 5 live that the Portuguese should not "ignore the concept" of going down.
Nevin also said that injuries could make it "even more difficult" to avoid being dragged into a "battle" to avoid the drop.
"You have to be realistic about relegation," he said. "Absolutely do not ignore it. When you lack confidence, that can spread through the team and it all can go against you.
"I don't think they'll go down, but don't bury your head in the sand. It's a possibility, so don't ignore it.
"I didn't think they'd be down there after 16 games. What about 20 games? What about 23 or 24? Then you start thinking about it."
Fabregas, speaking at a Facebook question and answer session, added: "Right now is not the time to think of where we might finish."
Source:BBC
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