That is one useful interpretation of the Beta distribution when it is used as a conjugate prior distribution to the binomial distribution. It breaks down a bit when you consider the possibility that it is perfectly legitimate for $\alpha + \beta < 1$ or even for $\alpha + \beta < 1/2$, meaning that $\alpha + \beta$ being the prior sample size is also only one possible interpretation of the parameters.
More generally, these are concentration parameters, which are a class of parameters that govern probability distributions over probability distributions. Concentration parameters have an interesting property. The smaller they are, the more sparse the distribution is. In the case of the Beta distribution, as $\alpha,\beta \rightarrow 0$, more and more of the probability is concentrated on the probability parameter $p$ being 0 or 1. Another interesting property of a concentration parameter is that when they all equal one, all possibilities are equally likely. Yet another property is that, as the concentration parameters get larger, the distributions tighten about the expectations.
This is one reason why it is sometimes useful to reparameterize the Beta distribution by one of its measures of central tendency (say its mean) and a dispersion parameter that governs the uncertainty in that mean. There are several ways to do this, up to and including expanding the Beta distribution model so that the mean and dispersion parameter are independent of one another (which is not the case for the two-parameter Beta distribution).
