The Waffen SS was an evolving organization, so its reliability and competence varied during the war.
Until 1942 the Waffen SS could not be seriously considered elite or even serious independent military units; as they were still primarily regiment-sized formations. 1st SS Division for instance was merely an attachment to the Heer's 17th Infantry Division in Poland. This was because the SS limited itself primarily to volunteers and those who fit Himmler's vision for the future Reich.
Serious expansion only began in 1942 - with the SS beginning to import experienced Heer officers instead of simply picking people for ideological purity. This expansion eventually resulted in the creation of the SS Panzer Corp consisting of the three most famous SS Panzer Divisions - Life Guard Adolf Hitler, Das Reich, and Totenkopf.
The SS Panzer Corp first saw serious action in early 1943, as part of Third Kharkov and Kursk, and in many ways their reputation was created as a result of these two battles. The SS Panzer Corps was rather instrumental at securing Manstein's victory at Third Kharkov - although it bears remembering they suffered very heavy losses that other Panzer Divisions would have found rather unnecessary. They also performed well at Kursk; arguably out-performing the Heer's Panzer Divisions for the first (and only) time.
After that however the SS degenerated into the Wermacht's shadow army. Himmler, seeking to secure more personal power, further expanded the SS (and circulated propaganda promoting them) until it eventually reached a staggering count of 30+ Divisions, the majority of which were Volkstrum (militia composed of old men and boys) or worse Ost battalion (Russian prisoners coerced to fight for the Germans) quality. The latter were particularly useless, unlike the earlier foreign volunteer Divisions like the Viking Division, as the soldiers were all unwilling and more liable to mutiny and shoot their own officers (with some units eventually doing so). Indeed Dennis Showalter, one of the leading experts on the German Army, once acidly noted that the only "German" component of some of these Divisions were the German shepard dogs.
Hence by 1944 it is difficult to consider the SS as being "elite" in any serious way. The SS Panzer Divisions sometimes performed well, but were also utterly crushed in other situations - just like all the other Panzer Divisions. From a strictly military perspective however they were just a needless duplication of resources - they could have been just regular Heer Panzer/Infantry Divisions and the German Armed forces as a whole would have worked better.
In summary, the SS could not be considered to be "elite" militarily except for a small period in 1943 when they did very well and imported some very good Heer officers; and much of their supposed prowess was a result of their key role at Third Kharkov + Kursk. However, after that they were expanded into an inconsistent shadow army that was raised primarily for Himmler's benefit; and their performance was rarely better than their Heer counterparts. The SS's insistence that they were still "Elite" in 1944 was mostly propaganda to justify the fact that their entire existence was more of a hindrance to the effective defense of Germany because of the duplication of the Heer command structure just to satisfy Himmler's ego and mistrust of the Heer.
On a side note, I have to say that the "best" and most elite German units often had very uninspiring names and hence are almost forgotten. The best single Panzer Division performance for instance was turned in by 11th Panzer Division at the Chir in 1943, where it basically fought an entire Soviet Tank Army on its own. Without 11th Panzer's actions there would not have been a Third Kharkov for the SS to even win. The highest-scoring armored battalions meanwhile were in fact several Sturmgetchutze Abteilung - "Assault Gun Battlions" - whose modest complement of Stugs killed more than any "Big Cat" Tiger or Panther battalion ever dreamed of approaching. The Second World War was not a war where the most glamorous units did most of the work.
The first question to ask is were they actually elite?
The Waffen SS was a very large organisation made up of many different divisions that were of different quality. However if you had to make a broad statement the SS went through different stages.
The first stage is in the early part of the war. Since the SS divisions were mobile infantry they performed very well within the doctrine of the German Armed forces at the time. Since they were often held back as reserves and deployed at crucial points in it lead to a reputation of elitism. Also at the same time the entry requirements to enter the SS were very high you had to prove your Aryan heritage back many generations.
At this point in time it would be fair to say the SS were elite after a sort they performed well in these early campaigns.
However as time wore on the SS became more of a German foreign legion this included people who only a few years early would have been considered inferior.
Also to note there were the Dirlewanger division of criminals in the SS so in that way there were several divisions that were no way elite.
To sum up some divisions were considered Elite mostly due to their "racial purity" however many divisions did perform well in the war.
The third Reich at War Richard Evans
SS A new history Adrian Weale
If you're interested, I've also answered a similar question in the past.