To reach the reduced carbon emission targets proposed by the Paris agreement, one of the widely proposed decarbonizing strategies, referred to as negative emissions technologies (NETs), is the production and combustion of bioenergy crops in conjunction with carbon capture and storage (BECCS). However, concerns have been increasingly raised that relying on the potential of BECCS to achieve negative emissions could result in delayed reductions in gross CO2 emissions, with consequent high risk of overshooting global temperature targets. We focus on two particular issues: the carbon efficiency and payback time of bioenergy use in BECCS and the potential constraints on the supply of bioenergy. The simplistic vision of BECCS is that 1 tonne of CO2 captured in the growth of biomass equates to 1 tonne of CO2 sequestered geologically, but this cannot be the case as CO2 is emitted by variable amounts during the lifecycle from crop establishment to sequestration below ground in geological formations. The deployment of BECCS is ultimately reliant on the availability of sufficient, sustainably sourced, biomass. The two most important factors determining this supply are land availability and land productivity. The upper bounds of the area estimates required correspond to more than the world's harvested land for cereal production. To achieve these estimates of biomass availability requires the rapid evolution of a multitude of technological, social, political and economic factors. Here, we question whether, because of the limited sustainable supply of biomass, BECCS should continue to be considered the dominant NET in IPCC and other scenarios achieving the Paris targets, or should it be deemed no longer fit for purpose?