If you move the 1s around to make this more "random", then you're bound to see some clusters of 1s, separated by longer stretches of 0s, with the occasional lone 1. However, if the process behind the bits is "totally random", then there is no particular reason why given 1s would cluster up. Yes, they come in groups, but not necessarily because of any hidden parameters.
It's interesting how shorter repetitions are considered lucky patterns, but a more distributed pattern like the one above is not usually noticed, unless you actually keep track over many days.
That is fine. But the intervals between clustered members should have a normal distribution if the apparent clustering is caused by randomeness. This discussion came up because many people have reported that blocks came up within hours after days or weeks when there were nothing. It;s these high sigma events appearing to happen too often that makes me suspect it's not random. I think rigorous measurement is needed to determine if it's truely random. Statistics arguements without data cut both ways.
A probably related phenomena is apparently slower machines get many more blocks than faster machines.