Understanding this pattern behind the Fibonacci sequence

The ratio between Fibonacci numbers soon settles down to a number close to $1.618$. This number is called the Golden Ratio.
You get an extra digit every time the Fibonacci numbers have increased by a factor 10.
$1.618^4=6.854$ and $1.618^5=11.09$
Once the ratio settles down, you get at least one extra digit every five numbers. Sometimes the extra digit arrives sooner, and you only get four numbers with so many digits.

Nice observation!

Here is an explanation:

The $n$-th Fibonacci number $F_n$ is asymptotically equal to $\varphi^n/\sqrt5$, where $\varphi=(1+\sqrt5)/2$. This implies that the number of digits in $F_n$, which is essentially $\log_{10} F_n$, is asymptotically equal to $n\,\log_{10}\varphi\approx0.2090\,n$. As a consequence, there are either $4$ or $5$ Fibonacci numbers with $d$ decimal digits, because $1/0.2090\approx4.785$.

[adapted from Wikipedia]