This paper discusses how the proximity to the L-H threshold affects the confinement of ELMy H-modes at high density. The largest reduction in confinement at high density is observed at the transition from the Type I to the Type III ELMy regime. At medium plasma triangularity, δ≅0.3 (where δ is the average triangularity at the separatrix), JET experiments show that by increasing the margin above the L-H threshold power and maintaining the edge temperature above the critical temperature for the transition to Type III ELMs, it is possible to avoid the degradation of the pedestal pressure with density, normally observed at lower power. As a result, the range of achievable densities (both in the core and in the pedestal) is increased. At high power above the L-H threshold power the core density was equal to the Greenwald limit with H97≅0.9. There is evidence that a mixed regime of Type I and Type II ELMs has been obtained at this intermediate triangularity, possibly as a result of this increase in density. At higher triangularity, δ≅0.5, the power required to achieve similar results is lower.