German Vice Chancellor Sigmar Gabriel said in an interview that Chancellor Angela Merkel's conservatives had "underestimated" the challenge of integrating record numbers of migrants.
Gabriel leads the Social Democrats (SPD) -- the junior coalition partner in Merkel's government -- and his comments come as campaigning kicks off for a federal election next year and regional elections in Berlin and the eastern state of Mecklenburg-Vorpommern.
Hundreds of thousands of migrants arrived in Germany from the Middle East, Africa and elsewhere last year. There is widespread concern about how to integrate them into German society and the labour market, and support for the anti-immigrant party Alternative for Germany (AfD) has grown.
"I, we always said that it's inconceivable for Germany to take in a million people every year," Gabriel said in an interview with broadcaster ZDF. "There is an upper limit to a country's integration ability," he added at a news conference on Sunday.
He said Germany had 300,000 new school children due to the migrant influx and added that the country could not manage to integrate so many into the school system every year because there would not be enough teachers.
Gabriel also criticized Merkel's catchphrase "Wir schaffen das" ("We can do this"), which she adopted during last year's migrant crisis.
The chancellor used the phrase at a news conference in late July after a spate of attacks on civilians in Germany, including two claimed by Islamic State, that put her open-door migrant policy in the spotlight and dented her popularity.
Merkel rejected Gabriel's criticism in an interview with broadcaster ARD and said the federal government had worked hard with state and municipal authorities to solve problems, changed laws and provided funding.
She suggested she was open to changes in the EU's planned quota system, which aims to transfer of about 160,000 asylum seekers from Greece and Italy to other EU states, but said it was important to find a solution that shared responsibility.
It would not be acceptable, Merkel said, for governments to say: "We don't generally want to have Muslims in our country."
Instead of taking in refugees en masse, some eastern European countries want to provide other forms of help - such as border guards for other EU countries - and have suggested that the European Commission could offset those contributions against its proposed distribution quota.