Sort of, depending on what you mean by random. The article seems to imply the uniform distribution on all 52! orders, which 7 shuffles won't get you. For example, you can't reverse a deck with 7 shuffles. There are bets you can put on a deck that's been shuffled 7 times that pay better than the same bet on a uniformly selected order.
Yeah, that phrase "truly random" in the wikipedia page is misleading. At least they do include in the references a link to Bayer and Diaconis' paper. What the paper says is that (for a certain mathematical model of the riffle shuffle, which experimentally seems to match actual results produced by human riffle shufflers) if you measure the total variation distance d between the probability distribution of the order you get after k shuffles and the uniform distribution on all 52! orders, you get d=1 for k=1,2,3,4 (i.e., maximally far in total variation distance from uniform), d=0.924 for k=5, and d drops off exponentially after that:
k 5 6 7 8 9 10
d 0.924 0.624 0.334 0.176 0.085 0.043
The total variation distance is an upper bound for the absolute difference in probability over all subsets of permutations.
In particular, we can consider the set:
U = {all permuations that have never been seen by a human}
The counting arguments in the article lead us to conclude that the uniform probability of U is very close to 1, i.e. almost all permuations have never been obtained by a human shuffle. The Bayer-Diaconis result implies that for certain types of shuffles the probability of ending up in U is at least
1 - (the total variation distance above)
So for 8 shuffles, we have a probablity of at least 82.4% of landing in U. This is considerably weaker than "every shuffle is unique".
Actually, I think lutorn's point is that most people (including him) do not do a full riffle shuffle when they set out to shuffle cards, thus preserving the order of segments of the cards.
But the shuffling model used in the Diaconis and Bayer paper that the reply referred to (the "7 shuffles to randomize" result) takes that into account. In particular, the shuffling model gives reasonably high probability to "imperfect" riffle shuffles that take several cards from the left pile and then several cards from the right pile.
Basically, you split the cards at random, and then you draw sequentially at random from the two piles with probability L/(L+R) versus R/(L+R) (where L is the number of cards remaining in the left pile), to assemble the new deck. This will allow shuffle sequences that take a run from the left and then a run from the right.
As people above have noted, of course the "randomization" of the deck is not perfect after 7 shuffles. But for lots of Markov processes, including this one, the distance of the shuffled deck to the uniform distribution on all decks tends to zero exponentially fast. So you get a quick change-over from "not random at all" to "very random". See table 3 of the paper.
Persi Diaconis (http://www-stat.stanford.edu/~cgates/PERSI/), to whom this result is partly due, is a legend in mathematical probability. How many math professors are bona fide magicians?