Motivated by the importance of magnetization dynamics in nanomagnets for the development and optimization of magnetic devices and sensors, we measured and modeled spin wave spectra in patterned elliptical nanomagnets. Ferromagnetic resonance spectra for multiple nanomagnets of Ni80Fe20, fabricated by electron-beam lithography to have nominal short-axes of 200 nm or 100 nm, were measured by use of heterodyne magneto-optical microwave microscopy. Scanning electron microscope images taken of the same nanomagnets were used to define element shapes for micromagnetic simulations. The measured spectra show significant differences between nominally identical nanomagnets, which could be only partially attributed to uncontrolled shape variations in the patterning process, as evidenced by the limited agreement between the measured and simulated spectra. Agreement between measurements and simulations was improved by including a zone of reduced magnetization and exchange at the edges of the nanomagnets in the simulations. Our results show that the reduction of shape variations between individual magnetic random-access memory elements can potentially improve their performance. However, unambiguous determination of materials parameters in nanomagnets based on analysis and modeling of spin wave spectra remains problematic.