In linear simultaneous equations (SE), nuisance parameters and weak identification (or weak instruments) severely complicate exact and asymptotic tests. A sizable literature has proposed test procedures which aim at being robust to these difficulties. We reconsider such problems in the context of a general framework which combines statistical perspectives on multivariate linear regression, dimensionality analysis, and bound tests in econometrics. We study how hypothesis tests in a standard simultaneous equations model can be viewed as tests of rank restrictions (or dimensionality) on a multivariate linear regression (MLR).We adopt a finite-sample perspective, and show that exact tests and bounds can be obtained in this framework without relying on identification assumptions. This eschews the need for local asymptotic approximations, such as drifting sequences or local-to-nonidentification assumptions. For the problem of testing subvectors of structural parameters in a linear SE model, the bounds proposed can be viewed as a refinement of a general bound given in Dufour (1997), and the finite-sample analogue of the identification-robust asymptotic bounds proposed by Guggenberger, Kleibergen, Mavroeidis and Chen (2012) in a LIML framework. Simulation experiments illustrate the usefulness of the bounds as well as tests against alternatives which are unrestricted by the structure.