I know that "The Great Game" was originally concerned with Russian-British competition in Central Asia, but my question is was this competition and British fear founded on anything? My understanding of Russian history at this point leads me to believe that the Tsarist empire was not in a position to try and undertake such a massive task. Perhaps I am mistaken about this, but I would like to see evidence supporting, or disproving, the British concerns.
I think everyone who has posted here (@ihtkwot, @mgb, @Russell) have all brought up important points. I agree with @ihtkwot and @Russell that Russia simply did not have the military capacity to challenge Britain's hold on India (a territory that the British would have vigorously defended). Also, Russia seemed far more interested in territories to the west and the burning desire to acquire a warm-water port on the Baltic and Black seas well explained by this commentator. While British concerns of a Russian military invasion of India are almost certainly unfounded (and as @mgb notes, rationality does not factor in irrational fears), Britian did have a much more legitimate fear of Russian economic and military influence of India, rather than outright domination.
There is a very good analysis by Nikki R. Keddie on the Russian/British dynamic in Afghanistan's neighbor to the west (also a neighbor of British Indian territory at the time), Iran in her book Modern Iran: Roots and Results of Revolution. While there is a far more likely possibility of Russian invasion of Iran, the usual sparring ground between Britain and Russia was not on direct military engagement or even proxy wars, but rather focuses on influence of the government over economic and military concessions.
Had Russia been successful in endeavors to dominate India's neighbors, they would have been in a position to influence actors within India. While this could have been a stepping stone to invasion, it is certainly a bit of stretch to worry about a grand invasion from a region as contested as it was (particularly later when German and American interests started getting involved). Given British interest in an area that often contained few exploitable natural resources or economic wealth, it seems quite clear that the empire was very interested in maintaining this border region to protect India from outside influence.
The Soviet Union's invasion of Afghanistan in the late 20th century has its roots in the "Great Game" of the 19th century. At stake then was the British domination of India.
Russia had long wanted a warm water port to the south. Her first choice was Constantinople (Istanbul), which met with British opposition e.g. in the Crimean war. Her second choice was Persia (Iran), and there was Soviet and British competition for influence in that country up to, and including the early part of World War II. Her third choice was "India," (including modern Pakistan, long part of British India). According to William L. Shirer in the "Rise and Fall of the Third Reich," one reason that the Soviet Union stayed in a pact with Hitler's Germany was because Hitler initially offered the Soviet Union a port in India (the offer was reneged when Germany invaded the Soviet Union).
So yes, Britain had reason to fear that a Russian penetration of Afghanistan would work against her interests in India, (modern) Pakistan, or Iran.
William Dalrymple, author of Return of a King: The Battle for Afghanistan 1839–42 seems to argue these fears were indeed ill-founded, at least in the sense that they led to an ill-founded strategy. This is from The Economist's book review:
Thus began the “Great Game”, an entirely unnecessary competition for Afghanistan between Russia and Britain, conjured up by armchair polemicists in London.
It's not entirely clear which author's view the sentence is meant to convey, but this interesting related interview makes me think it is broadly Dalrymple's, who is recognized as an expert author on the region's history.