In this work, we consider the problem of uncertainty estimation of the results obtained through theoretical calculations using a composite quantum chemistry scheme. Each component of the composite scheme carries its own individual uncertainty, originating principally from a finite size of the basis set used in the calculations. Of practical interest is the total uncertainty, i.e. the uncertainty of the sum of all components. We first show that the conventional error propagation rules that treat each component as a statistically uncorrelated variable are not well-justified in practice. To remedy this, we propose a method of estimating the total uncertainty of the composite scheme which does not rely on the assumption that the components are independent. It is formulated as a series of random walks with different starting values that represent the uncertainty of each component of the composite scheme. This method is first applied to two example composite schemes for the water dimer and the nitrogen molecule that illustrate its most salient features and allow for a deeper analysis. Next, the method is tested for a larger set of interaction energies from the S66 dataset for which trustworthy reference data are available and the error can be assessed unambiguously. It is shown that the method provides reliable and reasonably tight uncertainty estimates at an arbitrary predefined confidence level required in a given application. While the focus of the paper is on composite schemes, we believe that the general idea can be useful more broadly in computational chemistry and physics.