Recent studies have shown that on JET with the Be/W ITER-like wall (JET-ILW) in high beta discharges with high D2 gas rates the inter-ELM temperature pedestal growth is saturated half way through the ELM cycle, leading to plasmas with reduced confinement, and that the linear MHD stability of these pedestals is inconsistent with the Peeling-Ballooning paradigm. In this paper, the inter-ELM evolution of the edge current density is investigated in a wide range of type I ELMy H-modes on JET-ILW. It is found that in discharges at low gas rate the peak edge bootstrap current continuously increases until the ELM crash, while it saturates during the ELM cycle at high gas rates. The effect of current diffusion on the build-up of the edge current inter-ELM is assessed by simulating the Ohmic current contribution with the JETTO transport code. The simulations indicate that current diffusion contributes little to the time evolution of the total edge current in the second half of the ELM cycle and therefore does not explain why JET-ILW type I ELMy pedestals at high gas rate and high βN are found to be stable to Peeling-Ballooning modes.