The amyloid fibril formation by $\alpha$-synuclein is a hallmark of various neurodegenerative disorders, most notably Parkinson's disease. Epigallocatechin gallate (EGCG) has been reported to be an efficient aggregation inhibitor of numerous proteins, among them $\alpha$-synuclein. Here we show that this applies only to a small region of relevant parameter space and that under some conditions, EGCG can even accelerate α-synuclein amyloid fibril formation through facilitating its heterogeneous primary nucleation. Furthermore, we show through quantitative seeding experiments that contrary to previous reports, EGCG is not able to re-model α-synuclein amyloid fibrils into seeding-incompetent structures. Taken together, our results paint a complex picture of EGCG as a compound that can under some conditions inhibit the amyloid fibril formation of α-synuclein, but the inhibitory action is not robust against various relevant changes in experimental conditions. Our results are important for the development of strategies to identify and characterise promising amyloid inhibitors.