  The box-counting (BC) algorithm is applied to calculate fractal dimensions of
four fractal sets. The sets are contaminated with an additive noise with
amplitude $\gamma = 10^{-5} \div 10^{-1}$. The accuracy of calculated numerical
values of the fractal dimensions is analyzed as a function of $\gamma$ for
different sizes of the data sample ($n_{tot}$). In particular, it has been
found that a tiny ($10^{-5}$) addition of noise generates much larger (three
orders of magnitude) error of the calculated fractal exponents. Natural
saturation of the error for larger noise values prohibits the power-like
scaling. Moreover, the noise effect cannot be cured by taking larger data
samples.
