Let's look at raw purchasing power (Which, for the last time, is directly tied to inflation).
In 1950, the FMW was $.75/hour. That's about $7.20 in today's dollars. For that the average rent of $42/month could be earned in about 1.5 weeks' worth of work.
In 1960, the FMW was $1/hour or $7.81 in today's dollars. For that, the average rent of $71 could be paid off in just UNDER 2 weeks.
In 1970 (the golden age, so to speak) the FMW was $1.60 or $9.54. For that, the average rent of $108 could be paid off in about 67-68 hours.
1980, the FMW is $3.10 ($8.70 in today's dollars, higher than 1960). For that, the average rent of $243 would take just under 2 weeks to pay off (78 hours).
1990, FMW is $3.80 or $6.73 in today's dollars. Average rent of $447 takes 118 hours worth of work to pay off.
2000, FMW is $5.15 or $6.92 today. Average rent is $602. Almost the same 117 hours.
2012, FMW is $7.35 and the average rent is $789 taking 109 hours to pay.
The simple fact is that the minimum wage has been more or less keeping up with inflation (In the 70s, outpacing it) and the simple facts are that, in very short order, it gets priced out of being viable again.
You can argue a multitude of factors are in play but the one, undeniable fact is that the FMW has been keeping up with inflation (and, thus, purchasing power).
Why won't the MW get you a decent apartment anymore? I couldn't say. It could be that middle class incomes have been increasing faster than the minimum wage and the demand they've spurred has priced poor people out of the market.
Whatever the case might be, it's a 74 year old experiment that hasn't yielded any long-term results and I think it's time to explore other options.