A total of 3,271 such cases had been filed by the end of March, the Federal Office for Migration and Refugees said in a written reply to a question from a lawmaker.
Such complaints could be deemed valid if authorities are found to have failed to issue a decision on an application within a specific timeframe.
About a third of the cases are before the courts in Germany's most populous state, North-Rhine Westphalia.
Migration office chief Frank-Juergen Weise had admitted in February that the authorities were running a massive backlog of refugee applications, after 1.1 million asylum seekers arrived in Germany last year alone.
He said then that between 670,000 and 770,000 people who arrived in Germany last year still had not received the final decision on their asylum applications.
The interior ministry said at the end of April that the asylum application process lasts an average of 5.8 months, up from the 4.9 month average in last August.