The aggregation of proteins is considered to be the main cause of several neurodegenerative diseases. Despite much progress in amyloid research, the process of fibrillization is still not fully understood, which is one of the main reasons why there are still very few effective treatments available. When the aggregation of insulin, a model amyloidogenic protein, is tracked using thioflavin-T (ThT), an amyloid specific dye, there is an anomalous occurrence of double-sigmoidal aggregation kinetics. Such an event is likely related to the formation of ThT-positive intermediates, which may affect the outcome of both aggregation kinetic data, as well as final fibril structure. In this work we explore insulin fibrillization under conditions, where both normal and double-sigmoidal kinetics are observed and show that, despite their dye-binding properties and random occurrence, the ThT-positive intermediates do not significantly alter the overall aggregation process.