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Migrants who were forced off a train by German border police officers arrive at the train station of the southern German border town of Passau on Monday.Credit Markus Schreiber/Associated Press

BERLIN — After a full week of record arrivals, Munich appeared dangerously close on Saturday to missing its pledge to provide a bed for every migrant heading its way. The police appealed to the public for donations of blankets and camping mattresses. Cookies and water would be welcome, too.

On Saturday alone, 13,000 people, a new peak, pulled into Munich’s main train station. Munich’s mayor, Dieter Reiter, had for days been urging Chancellor Angela Merkel’s government and Germany’s 15 other states to shoulder their share of the burden — a mirror of the problem that Berlin faced at the European level, as it asked for a plan to distribute the arrivals fairly among the European Union member states.

For most of the newcomers, their arrival at the station was the long-anticipated end of a journey that began under harrowing conditions in the Middle East. But for the authorities in Munich — and for Ms. Merkel, as well — it was a tipping point.

The chancellor, who last week posed for photos with grateful migrants in Berlin, now faced anger from within her conservative bloc and from the leaders of most of Germany’s 16 states.

At the same time, in Europe, she was making no headway with arguments for a fair distribution of migrants ahead of a crucial meeting in Brussels on the issue Monday.

And one more problem was looming: In a week Oktoberfest would open its beer tents in Bavaria, drawing six million revelers, many whom would pass through the Munich train station. Officials feared the situation could spiral out of control.

As frustration mounted in Berlin, Munich, Budapest and Brussels, the chancellor did an about-face: After throwing open its borders a week earlier, Germany would reimpose controls on its open borders. Although not unprecedented, the step dealt a blow to European pride in freedom of movement — and signaled that Germany’s patience was spent.

Some of the pressure on Ms. Merkel had come from the governor of Bavaria, Horst Seehofer, who complained that she had opened the borders with very little advance notice and that his state was scrambling to accommodate a relentless flow.

European Refugee and Migrant Crisis

Mr. Seehofer, the conservative leader of Bavaria’s governing Christian Social Union, is nominally the chancellor’s ally in her center-right bloc. But the two are wildly dissimilar: She is the scientist raised in East Germany who has steered her party to the center, while he presides over the bloc’s deeply traditional Roman Catholics.

He likes to chide her, and now he feared for public security in Bavaria.

By midday Saturday, as the authorities in Munich, the Bavarian capital, faced a record number of arrivals, Mr. Reiter, the mayor, warned that the situation was becoming unbearable.

“This is too much, even for Munich,” he said. “I keep hearing from Berlin that there are talks about a way to distribute refugees, but we are not talking days or weeks, it’s got to happen right now.”

Anger had also reached a boiling point among leaders of other states, which in Germany not only must take in a certain quota of migrants, but are also required to pay for housing and educating them. The governors insisted that the six billion euros ($6.8 billion) Ms. Merkel had so far offered in assistance was woefully inadequate.

Under pressure, Ms. Merkel called on her interior minister, Thomas de Maizière, to announce the new border policy. It was cast as a signal to European partners that even wealthy Germany could not shoulder the burden of accommodating up to 800,000 migrants this year.

By Monday, Ms. Merkel’s deputy, Sigmar Gabriel, was arguing that in fact the country could face an even greater number of migrants in 2015. “There are many indications that in this year we will not see 800,000 refugees, as predicted, but a million,” Mr. Gabriel wrote in a letter to members of his center-left Social Democrats.

“Germany is strong and can handle a lot,” Mr. Gabriel wrote. “Nevertheless, in the past few days we have experienced how despite our best efforts, our abilities have reached their limits.”

Graphic Seeking a Fair Distribution of Migrants in Europe German and European Union leaders have called for European countries to share the burden of absorbing the hundreds of thousands of migrants who have poured into the continent this summer.