While most historiography is indeed in agreement that Nazi-Germany largely failed to penetrate the Allies, the counter-intelligence record of Nazi-Germany seems to point that they enjoyed some considerable successes, but also some major failures.

The counter-intelligence landscape of Germany between 1933 and 1945 was rather complex, with numerous reorganizations and multiple agencies overlapping and competing, some of which were formally not government but Nazi-Party agencies.

When the Nazi's seized power in 1933, the chief counter-intelligence agency at their disposal was the Abwehr, a military counter-intelligence agency. Under Wilhelm Canaris, they were quite succesful in destroying a number of foreign intelligence operations. Perhaps the best known case is Operation NORDPOL, a deception game against the British Special Operations Executive (SOE).

The SOE had intended to organize a sabotage network in the Occupied Netherlands in 1941, but one of the radio transmitters fell into the hands of the Abwehr. The Abwehr forced the operator to participate in deceiving the British, who continued to send him instructions until at least 1944. Dozens of Dutch resistance fighters that the UK sent across the Channel to carry out sabotage operations in the Netherlands were caught this way by the Abwehr and executed.

Another victory for the Abwehr was their uncovering over the Soviet GRU networks throughout Western Europe, known as Rote Kapelle. The Abwehr detected the network's existence, which spanned from Belgium and the Netherlands to Germany itself. Upon their discovery, the Gestapo - another counter-intelligence agency - swiftly moved in and arrested hundreds of its members in 1942.

At the same time the Abwehr, especially as the Nazi regime radicalized, became a hotbed of that typical Prussian bureaucratic resistance. Anti-Nazi sentiments existed within the Abwehr, and some of its officers participated in a coup attempt against Hitler. By 1944, Hitler fired its chief Canaris because unlike the rest of the Nazi Yes-men, he predicted that the war was lost. The Abwehr was integrated into the centralized RSHA that stood under the command of the SS.

The Gestapo was another counter-intelligence agency, but much more oriented towards the detection of domestic resistance groups. It was a secret police force, chiefly responsible for detecting political crimes, but its broad task naturally also included the detection of spies, saboteurs and foreign agents. The Gestapo was grossly understaffed for the job it was given however, having only a few thousand officers behind a desk to cover all of Germany with its 80 million citizens. In reality, the Gestapo was a very reactive bureau, acting on denunciations provided by citizens on their neighbours and colleagues.

Another major counter-intelligence agency was the SS' Sicherheitsdienst. Its chief of counter-espionage was Walter Schellenberg, probably the most succesful intelligence chief of Nazi Germany. His most well-known action went down in British history as the "Venlo Incident" of 1939, again in the Netherlands.

MI6 had been trying to establish contacts with the German resistance, and especially with people in the German military as MI6 was frantically trying to improve its intelligence position on the German military. Schellenberg and the SD seized on this opportunity to convince MI6 that there were a few people willing to meet at the Dutch-German border near the Dutch town of Venlo. Once the MI6 officers arrived in Venlo, they were seized by the SS together with a Dutch intelligence officer and abducted to Germany. The Dutchman died in a German hospital, the English officers pretty much spilled all of MI6's secrets under severe torture, and were finally liberated in 1945.

Schellenberg was personally rewarded and transferred to the SD's foreign intelligence section. The SD as a whole was merged into the larger RSHA headed by Reinhard Heydrich, who was assassinated in 1942.

It needs to be stressed here that despite some victories, German counter-intelligence was fighting an uphill battle. Being ever more encircled by enemies, with a leader who absolutely despised intelligence and considered it a waste of his time, and the fact that the US, UK and Soviet Union were intercepting and reading mass volumes of German communications - while German sigint produced less and less, the Allies could simply gather more intelligence than the Nazi's could ever hope to protect.

Especially the RSHA devoted considerable resources to the destruction of the Jews and genocide, being responsible for the Final Solution and organizing the Einsatzgruppen that hunted down Jews throughout occupied lands. To the Nazi's, this could not be seen separately from counter-intelligence work, just as the KGB did not see a difference between an American spy and a political dissident. Being convinced that the Jews controlled America for example, Jews anywhere in Europe constituted a dangerous fifth column that needed to be detected and arrested as quickly as possible.

Based on analysis of their reports, the Abwehr seems to have been the least affected by anti-Semitic bias, while the SD and RSHA were quite extreme in their belief in anti-Jewish conspiracy theories.

So to conclude, German counter-intelligence witnessed some success, but they too suffered from Hitler's management, politicization, and were basically fighting a battle they could never win, while dedicating ever more resources to the destruction of the Jews.

For further reading I could recommend:

Jefferson Adams, "Historical Dictionary of German Intelligence" (2009). Contains useful bibliography lists.

Heinz Höhne, "Hitler's Master Spy" (New York 1999), about Wilhelm Canaris. More recent biographies also exist.