The comments from the head of state, who has a largely ceremonial role but whose words carry moral authority, were set to fuel a raging debate about how so many newcomers can be integrated into German society.
"We want to help. We have a big heart. However, there is a limit to what we can do," President Joachim Gauck, a former rights activist in communist East Germany, said in a speech on Sunday evening.
"Our ability to take in people is limited, although we don't know yet where those limits are," he added.
Gauck appeared to be striking a more cautious tone than Chancellor Angela Merkel who has said Germany can cope with the record influx.
It came amid signs of tensions among asylum seekers, part of a wave of mass migration into the European Union from the Middle East and parts of Africa.
Fourteen people were injured on Sunday when a brawl broke out between two rival groups in a refugee centre in the western German town of Calden, police said. The refugees were being housed in tents set up on a disused airfield.
A union representing German police officers has suggested that refugees should be divided according to their religion to try to reduce the risk of conflict.
Merkel's popularity has dropped sharply over her handling of the refugee crisis, two polls showed at the weekend, indicating a shift in the mood in the Europe Union's most populous country.
The central government agreed last week to give its 16 regional states around 4 billion euros ($4.5 billion) next year to help cover the cost of looking after refugees.
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