Compressive testing of specimens taken from relatively thin composite plates is difficult, especially due to the occurrence of buckling. To prevent buckling, the central portion of the specimens used for the compression test has smaller dimensions, and the specimens can be guided along their entire length. For these reasons, optical methods, such as digital image correlation (DIC), cannot be used for the compression test and strain rosettes cannot be glued onto the samples to determine Poisson’s ratio. In this study, compression tests of a glass fiber-reinforced polymer (GFRP) were conducted using both the ASTM D695 (Boeing version) and a newly proposed method. The new method involves using special specimens that allow T-type rosettes to be bonded to determine Poisson’s ratio, whose value of 0.14 was thus determined. SEM images of the failure surfaces were presented and interpreted. A finite element analysis (FEA) of the specimens tested in compression is also presented. The first analyzed case considers the homogeneous and orthotropic composite, loaded with a uniformly distributed force. The normal stress in the central section of the specimen, determined with FEA, has an error of 6.52% compared to that determined experimentally. Additionally, the strain in the center of the strain gauge, determined with FEA, has an error of 4.76% compared to the measured one. In the second case studied with FEA, the sample is loaded with a quasi-concentrated force, which can move in the direction of the symmetry axes of the cross-section, to study the effect of the eccentricity of the compression force on the state of stress. It was shown that the eccentricity of the force has a great influence: the stress distribution in the section of the specimen becomes strongly non-uniform. For a force eccentricity of 0.4 mm in the direction of the OX axis, the minimum stress decreases by 53.7%, and the maximum stress increases by 55.4%. In order to analyze the influence of some manufacturing defects, two other cases were analyzed by FEA, in which it was assumed that the thicknesses of the outer resin layers were modified, making them asymmetrical. For this final FEA, the specimen was considered to be composed of laminates. These results demonstrate the special attention that must be paid to the centric application of force in compression testing.