I still haven't been able to get ahold of the book that the Wiki cites, which claims that the Germans remained confident in the security of the code even though it had been cracked. However, that's not to say that there was no one realizing something was up, so to speak.

The possibility that Allied intelligence agencies had been able to break into German cipher systems during the Second World War was of interest only to a small group of German naval officers who worked dur- ing the late 1940s on a history of the German U-boat war for the British Admiralty. In the war diaries of the Commander U-boats, to which they had access, they found many details about the successes of the German naval decryption service, the B-Dienst, against the British and Allied crypto systems between 1940 and 1943. In addition, however, they dis- covered that the Commander U-boats had suspected a break in the secu- rity of the German naval Enigma system, reasoning that only Allied interception of German radio signals to and from the U-boats could explain the destruction of the German surface supply system in the Atlantic in June 1941, and the accuracy of the Allied position estimates for U-boats in the spring of 1943. But until the early 1970s, the German officers' efforts to get information from the British experts about this matter ended in failure. Thus, for many years the literature attributed the crisis in the U-boat war in the second half of 1941 and especially the turn of the tide in May 1943 primarily to the success of Allied radar and shipborne high-frequency direction finding (HF/DF), which became pub- licly known only after the war.^^1

There is far more on the subject as well, which I was able to dig up through another source:

The fifth and last reason for German inferiority is the broadest: a greater reluctance to face reality. This reluctance, combined with the lack of irrefutable evidence for enemy cryptanalysis, largely kept the naval high command from conceding during several investigations that its cipher might have been broken. The officers found it difficult to admit to themselves, to their chiefs, and to Hitler that everything they had said and done was worthless and would have to be redone. The result was disaster. Of course, the Allies, too, sometimes engaged in wishful thinking. Conferences on possible compromises of systems sometimes decided, as the Germans did, that none had occurred, largely because the cryptologists did not want to go through all the work of instituting new systems - devising, manufacturing and distributing the new machines, training the personnel, and phasing the system into operation with the inevitable blunders that would call down the wrath of fighting admirals and generals. But conditions differed for the Allies. They were less crippled by arrogance than the Germans"' and so more open to improvement. SIGABA'S greater strength enabled the Americans to restore security in case of a compromise by simply replacing rotors; the Germans would have had to substitute a whole new system for the Enigma. Competition between Americans and Britons in a joint endeavour helped keep failure from being hidden for very long. Civilians headed important sections of codebreaking and intelligence agencies more frequently in Allied forces than in German; because they were less concerned about their military careers than the officers and officials who headed the corresponding German sections, these civilians admitted unpleasant facts to their superiors more readily. For all these reasons, the Allies seem to have faced reality more. When an American cipher machine went astray in France in 1944, the American army code agency worked day and night to rewire the rotors of other machines, thus making the missing machine useless to cryptanalysts. And when the Americans got wind of the German solution to the military attache code, they distributed a new system. So, as one American cryptologist has said, 'We never kidded ourselves. The Germans and Japanese did kid themselves.'^^2

It really does seem like both sides were reluctant to admit that their codes might've been broken at any point, but the Germans were especially reluctant. I wasn't a big fan of Kahn, however, because he writes in far too opinionated-sounding of a manner, so I checked with another source:

The heart of the Ultra Secret was a Ger- man cypher machine, called the "Enigma." The Enigma was an elaborate device, using a system of revolving drums which made its cypher virtually unbreaka- ble. Believing the Enigma secure from enemy code-breaking, the German high command used it throughout the war for the highest-level military communications ^^3

And finally, I stumbled across a gem of a source that addresses precisely what the Germans were saying about the possibility of their code being cracked:

In September 1942, British interrogators recorded the following exchange between two German prisoners of war, one a U-boat radio operator and the other a member of the Abwehr's commando unit, Lehrregiment Brandenburg, on the security of codes:

Radio operator: We have often cracked the British code, during the Norwegian campaign for example, but they will never crack the code we had in the Navy. It's absolutely impossible to crack.

Abwehr commando: Everyone says that of their own code.

Radio operator: What! They can't crack it.

Abvehlr commando: There's only one method that can't be deciphered and even that can be deciphered by expert mathematicians; I think they can break a code in the course of two years....

Radio operator: No, they can't crack it.

Abwehr commando: Oh, that's just one of those silly ideas people have.

Radio operator: No.'

It expands on that later, as well:

The German Navy's tested but reaffirmed faith in its codes, based on the "Enigma-M" machine,4 therefore commands par- ticular attention. "Our ciphers were checked and rechecked, to make sure they were unbreakable," recalled former Commander-in-Chief of Submarines (Befehlshaber der Unter- seeboote, or BdU) Grand Admiral Karl Doenitz, and each time they were rated "impossible for the enemy to decipher." So strong was this conviction that in 1970 the former director of German Navy signal intelligence refuted Kahn's findings while the 1974 ULTRA revelation failed to convince the former cryp- tographic specialist for the Naval War Staff (Seekriegsleitung, or Skl).^^4

Hope that answers your question!

^^1 Signal Intelligence and World War II: The Unfolding Story Jurgen Rohwer The Journal of Military History , Vol. 63, No. 4 (Oct., 1999) , pp. 939-951

^^2 Codebreaking in World Wars I and II: The Major Successes and Failures, Their Causes and Their Effects David Kahn The Historical Journal , Vol. 23, No. 3 (Sep., 1980) , pp. 617-639

^^3 Some Implications of ULTRA Roger J. Spiller Military Affairs , Vol. 40, No. 2 (Apr., 1976) , pp. 49-54

^^4 The German Navy Evaluates Its Cryptographic Security, October 1941 Timothy Mulligan Military Affairs , Vol. 49, No. 2 (Apr., 1985) , pp. 75-79