While this might be a bit of a stretch since in the end it is billed as an Allied military victory, I think the Battle of Anzio and its after effects failed to accomplish their primary objectives.
At the end of 1943/beginning of 1944 the Allies were bogged down assaulting the Gustav line, a series of German defensive works that spanned the Italian peninsula west to east, with its anchor in the town of Monte Casino. Churchill's idea was to land two infantry divisions to the northwest of the Gustav line near the city of Anzio. According to Rick Attkinson's book The Day of Battle: The War in Sicily and Italy, 1943–1944 this attack appeared to demonstrate two benefits, it would either force the Germans to divert troops from the Gustav line, making an Allied breakthrough there more likely, or the troops at the Anzio beachhead could advance inland and trap the defenders between several Allied Divisions.
The first landings took place in January 22, 1944, and at first things looked great. The Allies managed to land 36,000 soldiers and 3,200 vehicles without any casualties at all and the American General in command at the beach, John Lucas, quickly consolidated the beach head. Unfortunately, after this, things just got worse and worse for the Allies. With only two infantry divisions and no supporting armor, it was important for Lucas to advance rapidly inland, as the Anzio beachhead was surrounded by high ground, perfect for the defensive mastermind Albert Kesselring, who commanded the German forces in Italy. According to Lloyd Clark’s book Anzio: The Friction of War. Italy and the Battle for Rome 1944. Lucas considered his force too small for his mission, and worried about Kesselring’s inevitable counter-attacks that would feature heavy artillery and tanks, he also declined the aid offered to him by Italian partisans, who claimed they could help his divisions navigate the local, hilly terrain.
Thus, Lucas spent too much time consolidating the Anzio beach head, and Kesselring could move minimal reinforcements to the hills and mountains surrounding it, and a long battle of attrition that resembled the stalemate at Monte Casino grinded on for about four months. In the end, the Allies took the town and Abbey of Monte Casino after assaulting it directly 5 times, and it was only after Allied armies began marching north that Major General Lucian Truscott, who had replaced Lucas coordinated a successful breakout of the beachhead. Clark’s book also points out that in the aftermath of Anzio yet another Allied failure shows it head; as Truscott was driving east to capture retreating German division from the Gustav line, his commander, Lieutenant General Mark Clark ordered Truscott’s corps northward to liberate Rome, in what is widely considered a purely symbolic victory for the Allied armies. This decision by Clark allowed thousands of German soldiers and their equipment to withdraw to the Gothic line, another set of defenses similar to the Gustav line that was towards the north of Italy.
Thus we see not only did Anzio not directly lead to the Allied breach of the Gustav line (you could even argue the troops who breached the Gustav line were the ones that allowed the Anzio breakout), but the Allied divisions that landed failed to trap retreating German forces. So despite the fact that the Allies eventually “won” the battle of Anzio, the poor planning and decisions made would help the Germans keep the fight up in Italy until almost the last days of the war in Europe.
Sources:
The Day of Battle: The War in Sicily and Italy, 1943–1944; Rick Attkinson
Anzio: The Friction of War. Italy and the Battle for Rome 1944; Lloyd Clark
A non-exhaustive list:
Norway 1940: the British and French fail to hold Norway against a fairly light Wehrmacht force, despite maintaining a hold on the important port of Narvik for some time.
Greece and Crete 1941: stripping forces from the armies that had just trounced, but not finished off, the Italians in North Africa, Wavell and Churchill attempt to defend Greece from Italian and German attack. The defence is a farce against strong German mobile- and air forces and an undignified evacuation ensues. The Germans then attack the island of Crete with an air-mobile force. Allied mistakes and weakened air power (because of losses associated with Greece and the refusal of the British high command to allow quality Fighter Command units to leave Britain) see this very close-run battle turn to victory for Germany. The German force did however suffer heavily though in the ensuing evacuation the Commonwealth army and naval units suffered heavily as well. This represented a regrettable waste of forces that could have been useful in...
Malaya/Singapore/Burma 1941-2: A large Commonwealth force, given time to prepare and what should have been ample resources, is trounced out of the Malay peninsula by a much smaller Japanese force. Two British capital ships, which could have stopped the seaborne component of the attack and against which the Japanese had no available comparable ships, are sunk. The highly defensible island of Singapore is easily accessed by the Japanese army. This defeat is probably Britain's greatest ever, and represented the end of British, and indeed European, power in Asia forever. The Japanese force rolled on into Burma where they defeated the Commonwealth armies present, stiffened with forces sent in response to the initial attack. A counteroffensive in the Arakan Peninsula in 1942 made no progress whatsoever, took heavy casualties, and the British forces had to evacuate precipitously to avoid being encircled.
Philippines 1941-2: While the British Commonwealth forces in Asia were suffering their calvary, US Forces under Douglas MacArthur were attempting to defend the Philippines. Warned, prepared and well-equipped (like the British, this refers to resourcing in general, in both cases the forces were not balanced and much equipment was not good enough), MacArthur's inept leadership, bad luck and the skill and detailed planning of the Japanese saw the US-Filipino force routed from their best positions and driven down the Bataan Peninsula. MacArthur's propaganda machine was so effective in making ordinary Americans believe in his skill and tenacity that Roosevelt was unable to slate to him responsibility for ultimate failure (FDR was probably also happy to have MacArthur 'busy' in the Pacific, rather than plotting against him at home).
The Channel Dash, 1941: after a raiding cruise in the Atlantic, three German heavy ships retired to the French port of Brest, where they were held by a British blockade. Here, they were bombed repeatedly by British aircraft and although Bomber Command was not very effective in 1941, occasional hits were obtained. In early 1941 the Germans executed a plan to sail the ships straight up the English Channel to safe harbours in northern Germany. Due to good luck and British ineptitude, all three ships pulled off the escape without suffering a scratch.
Dieppe 1942: A mostly Canadian force was landed on the French coast at Dieppe in 1942 for a sort of large raid. The idea was to test attack and the German defence of a coastal port, expand on the commando raid technique, and probably to placate the Soviets' demand for a Second Front. Co-ordination failures, rapid German response, and a multitude of small failures (for instance many tanks simply couldn't advance on the beach, which consisted of small, smooth stones) led to something of a disaster. The sheer difficulty of the mission was known, and was also a factor. Many of the lessons were applied to the D-Day landings.
The Dodecanese Campaign, 1944: Churchill was obsessed with Greece, and was the impetus behind this plan which landed Allied troops on Greek Islands. The British command failed to rapidly secure Italian co-operation, losing the chance of gaining support from the large Italian garrison. Also, the US government refused to take part, denying the support of American units including, crucially, long-range fighter aircraft. Out of range of meaningful support, especially air support, and with other higher priorities the Germans were able to marshall strong forces including paratroopers with lavish support by otherwise-obsolete Stuka dive bombers, to smash the British (and Greek loyalist) units, which had no hope of evacuation. The famous, elite Long Range Desert Group was squandered in this operation.
The Battle of the Java Sea. A combined American, British, Dutch, and Australian fleet (ABDA for short) under Dutch command attacked the Imperial Japanese Navy during Japan's invasion of the Dutch East Indies (now Indonesia) in February 1942. The ABDA fleet was poorly coordinated and the crews were generally exhausted. The Japanese force they were attacking was very cohesive, with the crews well-rested. The Japanese ships also outgunned the ABDA fleet. During the subsequent battle, the ABDA commander Adm. Karel Doorman was killed, along with over 2,000 sailors. ABDA lost 5 out of 14 ships with a 6th badly damaged. The Japanese lost no ships, although one destroyer did take some damage. About all the battle did was give the defenders of Java, along with a few other islands, a brief respite before the Japanese got there and overran the defenses anyway. The Japanese held Indonesia until the end of the war.