Okay, rolled a random civ in my multiplayer (Ice Age map, standard size, Prince) today - and just by chance I happened to be the Huns. The game's now most of the way through and I'm score leader having taken many cities early. This is my experience so far:
I had an ideal starting location - a large island with me, two other civs (both AIs) and two city-states. Not sure what game speed, but everything was cheaper than I"m used to in single-player (scout produced in 3 turns, teching very fast), which clearly benefits the early rush a lot and was something I hadn't factored in.
My army came together very quickly - by the time I had two rams and two HAs, as well as my faithful Warrior and the Honor opener GG, I was on the march towards Washington. Before turn 60 the island was mine, with the Hunnic cities of Washington, Edinburgh, Warsaw and Lhasa (and soon after a couple of newly-settled cities, as a lot of space was unused and there were exploitable resources and good production sites). This felt overpowered while I was doing it - but this was on Prince, against AIs that weren't garrisoning their cities that early and had no defenders (Washington had a spearman in a position to attack an HA but never used it; the cities themselves kept firing at HAs rather than rams), and I haven't tried similar rushes with other early-game civs for comparison.
Needing to develop Optics before continuing my rampage stalled me, but it turned out the closest landmass had only three cities: Monaco, Zanzibar and Tyre. As these were AIs and Tyre and Zanzibar were both mercantile, I decided to continue the rush. HAs were starting to lose effectiveness, but I eventually took Monaco and, within my new borders, could upgrade the rams to trebuchets (which of course left me light on melee units to actually take cities). The rush was definitely over after Zanzibar fell - in fact I ended up losing my army (including my faithful Warrior, now a Longswordsman, and an almost fully-promoted HA) in the abortive attack on Tyre, which only fell after I rebuilt and supported my attack with Privateers.
For a start I overextended. The early units were punishing on my economy, so I gained gold slowly (and soon ran out of nearby city-states to bully - I should have kept one or two as my banks), which meant late courthouses (I still have a couple of puppeted cities in the late 18th Century - the badly-placed Monaco, and I think one other), and struggles with unhappiness as well as long puppet periods hampered growth for the cities without wheat or sea resources (everywhere else was plains, desert or tundra, and I have only one river in my territory). Warsaw was my third city and is still pop 3; I can't usefully maintain it at any higher level (it does have to devote one pop to Mt Fuji, admittedly).
Overall, my experience has been much what I'd have expected and what I've argued, and as others have reported. In this case the Hun golden age lasted longer than I'd expected (into the early Renaissance), and having so many early cities - even if some were low-pop - kept me in the lead technologically for a while. But I now find myself in the lead scorewise (by about 150 points) in the 18th Century, yet without any obvious win condition I can go for. I lost out on the science wonders and have fallen behind technologically; I'm generating respectable culture but far too little to transition to that victory. City-states are naturally wary around me, although not yet at a point where I can't bribe them back - still I'm too far behind technologically to have a realistic shot at getting the UN. And domination is going to be hampered when up against two technologically superior human players at a point when none of the Hunnic advantages (other than a production bonus that very slightly helps Attila's Court - already a fantastic production city - to have marginally higher production) give me any edge in warfare. (EDIT: Oh yes, and this is the point where it's probably worth mentioning that one of said human players is Japan).
This is given a pretty much perfect start for the Huns in multiplayer, where I started the game paired with exactly enough cities for my early rush to take, all controlled by AI opponents who wouldn't be able to do anything to stall my early advance, and plenty of time (and a generally peaceful player group) to capitalise on my early captures. Certainly I could have played better in terms of capitalising on that early advantage, but even there - against other humans, and in a situation where victory over the AI would be automatic (Harun, the surviving AI, is trailing the three of us in everything and is around 300 points behind the weakest human player) - early Hun games really don't translate into a strong Hun late game, let alone a victory.