Possible, yes. Likely, I don't think so.
What were the Nazis after, in the first place?
We have to keep in mind the strategic goal of Nazi Germany: "Lebensraum", access to natural ressources not found on German soil, and a better geostrategic position on Europe's map.
As such, it is a common fallacy to say "if they only hadn't attacked Russia". Because Russia was the whole point of Nazi Germany's grab for power. It had an abundance of arable land, and resources in the form of oil and ores. The communist movement was the greatest threat to the Nazis, politically speaking. Geostrategically, it was easier conceivable to remove the threat of Russia from one side than to defeat France and Britain on the other side of Germany. (Especially in the light of the outcome of the first world war.)
To point that out specifically, Germany had little to win by conquering France and Britain. The wealth of those countries was in oversea colonies, and Germany just did not have naval power to make a grab for those as well.
Russia's wealth, by comparison, seemed to be within reach.
So, what was it that the leaders of Nazi Germany envisioned as their mid-term goal?
The military goal for Operation Barbarossa was the Arkhangelsk-Astrakhan line, gutting Russia's industrial capacity and putting the new border on easily defendable natural borders like the Volga river.
It is important to understand the motivation for Germany's early moves in the war. Annexing Austria, Czechoslovakia, and Poland were directly building up to Operation Barbarossa. The invasion of Poland was answered by a declaration of war by France and Britain -- something Germany could well have done without.
The resulting invasion of Denmark and Norway was done to protect the supply of Swedish iron ore, which Germany was dependent on.
The attack on France was to remove the direct threat to Germany's western border. When France capitulated, that threat was removed. But Britain could not be convinced to withdraw from the war, not by awkward diplomacy (Ribbentrop...), not by force (Battle of Britain and the threat of Operation Sea Lion), and this is where Germany's strategy started to fall apart. Additional distractions showed up, like the North Africa campaign (initiated by Italy, and again against Britain which the Nazis had hoped to keep out of the war entirely), Crete, and Greece.
But this was all strictly a side-show. All these operations were merely for general protection of the strategic position. The whole point of Germany's warfare was to defeat Russia.
So, was it ever possible for Germany to win World War II?
I will not talk about "likely", just about "possible", and to that the answer is YES.
Getting Britain out of the war, as early as possible, would have been a major success. It would have freed up significant resources -- Luftwaffe units not lost in the Battle of Britain, paratrooper forces not blunted by the invasion of Crete, ground forces not tied down in Greece and North Africa. Plus, it would likely have meant keeping the USA out of the European war as well.
What would it have taken to achieve this? There is a variety of ways it might have happened:
- Better diplomacy. Ribbentrop was a complete dud as Ambassador to the United Kingdom. Churchill was as steadfast an enemy as Germany could get. Things might have gone better here. (Though, as commentators pointed out, Germany had little to bargain with here. What little they had, together with military threats, could still have been delivered more competently.)
- Enigma not being broken. That would actually have been rather easy -- the invention of the reflector by Willi Korn increased the key space significantly, but also weakened the cipher as no character could be encoded to itself. Adding another rotor or two instead would have made Enigma much more resilient to cryptanalysis.
- U-boat warfare. German submarines sank significant amounts of crucial supplies, to the point where Britain considered those losses non-sustainable for some time. Some more success here (e.g. through better recon / air support) could have been a factor into tipping the balance.
- Battle of Britain. While Operation Sea Lion was pretty much a joke when we look at it today, the threat of it was very much felt by Britain at the time. Having lost signifcant forces in France, Britain felt very ill-prepared to face an invasion (Churchill's brilliant rhetorics nonwithstanding). With a better-prepared Luftwaffe, and the focus of attacks remaining on airfields, radar installations etc. (instead of shifting to bombing London as they did), that threat might have been made even more immediate.
All that is not even going into what could have been done different on the Eastern Front -- troops actually being prepared for winter warfare, Wehrmacht actually pressing for Moscow instead of repeatedly trying to reach a decisive encirclement of Russian forces in the open plains, things like that.
The key would have been to force Britain into an armistice. Or perhaps even persuading them into a pact with the devil before Poland was invaded. Again, I don't say it was likely, but it was possible, yes. Without Britain (and, by extension, the USA), Germany might have reached the A-A line, and it is hard to see Russia coming back from that.
All that being said, of course we would likely be looking at subsequent wars in decades to come. The Nazi philosophy was murderous -- Hitler was very plain as early as 1925 ("Mein Kampf") that he would not stand for any neighbouring country even capable of threatening Germany. So conflict with Britain would have been pretty much inevitable.
I shudder to think how that might have played out with Germany in secure possession of Russian and Romanian oil fields and ore mines...