As a paradigm of data processing, granular computation concerns processing complex data entities called granules, which arise from data abstraction and derivation of knowledge from data. This paper addresses granular computation within the framework of interval time series forecasting and evolving intelligent systems. It develops a generalized interval evolving possibilistic fuzzy modeling algorithm as an analytics tool capable to process interval data stream and to produce interval forecasts. The algorithm uses interval arithmetic in its processing steps, employs the notion of data density to adapt the current forecasting model as data are input, and computes (dis)similarity between interval data using the Hausdorff distance. Computational experiments include forecasting of an interval time series data produced by a synthetic time-varying model with parameter drift, and forecasting of financial interval time series using actual daily minimum and maximum values of the US and Brazilian main equity indexes, S&P 500 and IBOVESPA, respectively. The results suggest that the generalized interval evolving possibilistic fuzzy algorithm is highly effective to model and forecast interval time series. It has comparable or better performance than alternative evolving fuzzy and benchmark interval-based approaches.