Absence of proof is not proof of absence and Jeff Stallings (the source of John Cavan's answer) seems to have a lot of experience, but I'd still like to present some seemingly dissenting information.
First though, as a quick Google search will tell you, there is certainly no shortage of websites from trainers and organisations talking about littermate syndrome, including Ian Dunbar.
However, all of those articles use eerily similar structure and language and none of them cite any sources whatsoever other than personal experience. Also, at the time of me writing this, there exists no Wikipedia page on the subject. Certainly not a reason to discount it, but strange nonetheless for such a supposedly well-established phenomena.
More importantly, though, there is a piece in John Bradshaw's excellent "In Defence of Dogs" about an actual scientific study on just how much dogs bond to humans versus other dogs that seems to flat-out contradict the littermate syndrome theory.
[...] but there is one study that conclusively shows that dogs are indeed prone to bonding more strongly with people than with other dogs.
The subjects of the study were eight seven-to-nine-year-old mogrels that had been living as littermate pairs in kennels since they were eight weeks old; all had been fully socialized to people, and they were being looked after by one carer who was, as far as they were concerned, equivalent to their 'owner'. When the experiment began, the kennelmates had not been apart even for a minute during the previous two years, and hardly ever during their whole lifetimes. However, when once of each pair was taken out of earshot for four hours, the remaining dog's behaviour did not alter appreciably. Puppies separated from their littermates will usually yelp until they are reunited, but these adult dogs barely even barked. Moreover, the level of the stress hormone cortisol in their blood did not change as a result of the separation, provided the dogs had been left in their familiar pen. Overall, therefore, there was no indication that any of these dogs was upset, this despite the fact that, since they had virtually no history of being left alone, they would not have been sure that they would be back with their pen-mate in a few hours' time.
In contrast, when the dogs were taken to an unfamiliar kennel, they did become upset. They were visibly agitated, and their levels of stress hormone went up by over 50 per cent. Remarkably, this proved true whether they were on their own or with their kennelmate. When the two were together, they did not interact with one another any more frequently than usual; whatever the bond between them, it was not sufficiently comforting of confidence-building to help them cope with being somewhere new, outside their familiar territory. However, if their carer sat quietly with each dog in the novel kennel, it would stay near him and pester him for contact (which he responded to by brief episodes of stroking). This was apparently enough to alleviate the dogs' stress completely, because if the carer was there, their cortisol levels stayed close to normal.
These dogs, although they had kept the company of another dog their whole lives, behaved as if they were much more attached to their carer than to their brother or sister. While they had not led quite the same kind of life as a pet does, everyday experience suggests that the same if probably true of pet dogs.
 David Tuber et al., 'Behavioural and gluococorticoid responses of adult domestic dogs (Canis familiaris) to companionship and social separation', Journal of Comparative Psychology, 110 (1996), pp. 103--8.
 Subsequent research has shown that dogs' stress hormone levels are different depending not only on the gender of their owners or carers (lower if they are women) but also their personalities (lower if the owners are extroverts).
John Bradshaw's phrasing never makes quite clear whether these dogs are actually part of the same litter, referring to them as "living as littermate pairs", "kennelmates", "pen-mate", "brother or sister". Unfortunately, I don't have access to the original article and can't find a preprint somewhere either, but many of the websites on littermate syndrome also explicitly state that it can also affect unrelated puppies if you get them at the same time.
Does this discount littermate syndrome? Probably not, as the dogs in question were 7-9 years old when tested. Like I said, I'm also not discounting all these trainers' and organisations' personal experience, but it, at the very least, seems to be an area with basically no research into it whatsoever, at least none I could find easily, and, given Tuber et al.'s study, might well be something that only affects puppies.
Even if that's the case, however, it does not invalidate the fact that two puppies are not simply double the work compared to one puppy but a lot more, so all of John Cavan's caveats still apply.