It has become popular recently to apply the multifractal formalism of statistical physics (scaling analysis of structure functions and f(a) singularity spectrum analysis) to financial data. The outcome of such studies is a nonlinear shape of the structure function and a nontrivial behavior of the spectrum. Eventually, this literature has moved from basic data analysis to estimation of particular variants of multi-fractal models for asset returns via fitting of the empirical t(q) and f(a) functions. Here, we reinvestigate earlier claims of multi-fractality using four long time series of important financial markets. Taking the recently proposed multi-fractal models of asset returns as our starting point, we show that the typical ?scaling estimators? used in the physics literature are unable to distinguish between spurious and ?real? multi-scaling of financial data. Designing explicit tests for multi-scaling, we can in no case reject the null hypothesis that the apparent curvature of both the scaling function and the Hölder spectrum are spuriously generated by the particular fattailed distribution of innovations characterizing financial data. Given the well-known overwhelming evidence in favor of different degrees of long-term dependence in the powers of returns, we interpret this inability to reject the null hypothesis of multi-scaling as a lack of discriminatory power of the standard approach rather than as a true rejection of multi-scaling in financial data. However, the complete ?failure? of the multi-fractal apparatus in this setting also raises the question whether results in other areas (like geophysics) suffer from similar short-comings of the traditional methodology.