Proponents of a basic income (BI) claim that it could bring significant reductions in financial poverty, on top of many other benefits, including greatly reduced administrative complexity and cost. Using microsimulation analysis in a comparative two-country setting, we show that the potential poverty-reducing impact of BI strongly depends on exactly how and where it is implemented. Implementing a BI requires far more choices than advocates seem to realize. The level at which the BI is set matters, but its exact specification matters even more. Which parts of the existing tax-benefit system are maintained, and which parts are abolished, modified or replaced? The impact of a BI, be it a low or a high one, thus strongly depends on the characteristics of the system that it is (partially) replacing or complementing, as well as the socio-economic context in which it is introduced. Some versions of BI could potentially help to reduce poverty but always at a significant cost and with substantial sections of the population incurring significant losses, which matters for political feasibility. A partial basic income complementing existing provisions appears to make more potential sense than a full basic income replacing them. The simplicity of BI, however, tends to be vastly overstated.