Partitioned approaches for fluid-structure interaction generally involve solving the fluid and solid subsystems sequentially. This approach allows for modularity, thereby leveraging advanced commercial and open-source software capabilities to offer increased flexibility for diverse FSI applications. Most partitioned FSI schemes apply Dirichlet--Neumann partitioning of the interface conditions, in which the fluid transfers tractions to the solid subsystem, and the solid in turn transfers displacements to the fluid. Dirichlet--Neumann coupling has proven adequate in a wide range of applications. However, this coupling scheme is sensitive to the added-mass effect, and it is susceptible to the incompressibility dilemma, i.e. it completely fails for FSI problems in which the fluid is incompressible and furnished with Dirichlet boundary conditions on the part of its boundary complementary to the interface. In this contribution, we demonstrate that if the fluid is incompressible and the fluid domain is nearly-closed, in the sense that the fluid domain is furnished with Dirichlet conditions except for a permeable part of the boundary where a Robin-type condition holds, then the Dirichlet--Neumann partitioned approach is sensitive to the flow resistance at the permeable part and, in particular, convergence of the partitioned approach deteriorates as the flow resistance increases. The Dirichlet--Neumann partitioned approach then becomes arbitrarily unstable in the limit of vanishing permeability, i.e., if the flow resistance passes to infinity. Based on a simple leaky-piston model problem, we establish that in the nearly-closed case, the convergence rate of the Dirichlet--Neumann partitioned method depends on a so-called added-damping effect. The presented analysis provides insights that can be leveraged to improve the robustness and efficiency of partitioned approaches for FSI problems involving contact, such as valve opening/closing applications. In addition, the results elucidate the incompressibility dilemma as a formal limit of the added-damping effect passing to infinity, and the corresponding challenges related to FSI problems with nearly closed fluid-domain configurations. Based on numerical experiments, we consider the generalization of the results of the simple model problem to more complex nearly-closed FSI problems.