The aim of this work is to provide a mathematical contribution to explain the numerical instabilities encountered under certain combinations of physical parameters in the simulation of fluid-structure interaction (FSI) when using loosely coupled time advancing schemes. It is also shown how the same combinations of parameters lead, in the case of strongly coupled schemes, to problems that demand a greater computational effort to be solved, requiring for example a high number of subiterations. The application that we have in mind is FSI simulation for blood flow in large human arteries, but the discussion applies as well to several FSI problems in which an incompressible fluid interacts with a thin elastic structure. To carry out the mathematical analysis, we consider a simplified model representing the interaction between a potential fluid and a linear elastic thin tube. Despite its simplicity, this model reproduces propagation phenomena and takes into account the added-mass effect of the fluid on the structure, which is known to be source of numerical difficulties. This allows to draw conclusions that apply to more realistic problems, as well.