We model the Monte Carlo simulation of a simple associating system to understand and quantify the ability of the simulation to provide measurements of the energy and heat capacity. The simulation is examined for its convergence to the equilibrium degree of association, and separately for the precision of its measurements. The molecular model used for the study was proposed by van Roij; it is very simple, and allows the formation of only chain-like association structures while exhibiting only association and no repulsion or dispersion interactions. However, the model captures the essential features that make the simulation of associating systems difficult, and it yields to various types of analysis. Unbiased and biased simulation methods are studied. Modeling is facilitated by coarse-graining the system in terms of the total number of association bonds. A bond-balance kinetic model is used to examine the convergence behavior of the simulation. An unexpected finding is that the strength of association is the sole determinant of the rate of convergence, and that the size (volume) of the association sites does not separately impact the convergence rate. The precision of the calculation is quantified via a variance that is obtained from a Markov model of the simulation. We present contours that quantify the precision to be expected in the energy and the heat capacity, from which one can estimate the confidence limits to be expected when simulating a particular system at a given state with a given number of Monte Carlo trials. The results apply rigorously only to the model used in the study, but they should prove useful in gauging the difficulty to be expected in simulating realistic model systems. The biasing Monte Carlo algorithm offers dramatic improvement in both convergence and precision when simulating strongly associating systems, but it is not particularly helpful for weakly associating systems. The analysis also uncovers a finite-size effect that is manifested when measuring the heat capacity in strongly associating systems.