Well...no. Theoretically, if interest is 100% paid continuously throughout the year, you'll double your money in about .72 years. The issue is that we generally quote interest rates as annualized rates _after_ taking into account the compounding frequency (this is the difference between "APY" and "APR"), so the rule is usually applied to annualized rates, in which case it's correct to say that the error increases as the rates increase.
But suppose rates were actually quoted as the annual rate before compounding frequency is accounted for.
With monthly compounding, your return is
> (1 + r/12)^(12n)
where r is the annual rate and n is the number of years.
With weekly compounding, your return is
> (1 + r/52)^(52n)
Daily:
> (1 + r/365.25) ^ (365.25n)
More generally, if p is the number of interest periods in the year
> (1 + r/p) ^ pn
Taking the limit of p as it tends to infinity (ie "continuous compounding) it can be shown than the rate is
> e^rn
Some text books write this as "Pe^rt", where P is your starting principle and t is the number of years. Same same. Anyway...
To double your money, solve the following for n:
> e^rn = 2
> n = ln(2) / r
> n = ~.69 / r
Thus the rule of ln(2) is actually perfectly accurate if you're looking at an annual interest rate that is paid continuously, or close to it. Round ln(2) to 72 for simplicity and it's still pretty close. The big error bar comes from the lack of continuous compounding, which is what forces the hand-waving in the articles derivation ("for small R...").
P.S. Written off-the-cuff, it's been years since I looked at the math for this and I might be totally way off. This is not investment advice, this is hacker news and I'm positive some kind soul will correct my mistakes. After all, the best way to get correct information on the internet is to post something wrong on a forum of smart pedantics.