Carbon nanotubes (CNTs) are used in applications ranging from electrical engineering to medical device manufacturing. It is well known that the addition of nanotubes can influence the mechanical properties of various industrial materials, including plastics. Electrospinning is a popular method for fabricating nanomaterials, widely suggested for polymer scaffold manufacturing. In this study, we aimed to describe the influence of single-walled carbon nanotube (SWCNT) suspensions on polymeric poured films and electrospun scaffolds and to investigate their structural and mechanical properties obtained from various compositions. To obtain films and electrospun scaffolds of 8 mm diameter, we used poly-ε-caprolactone (PCL) and poly(cyclohexene carbonate) (PCHC) solutions containing several mass fractions of SWCNT. The samples were characterized using tensile tests, atomic force and scanning electronic microscopy (AFM and SEM). All the studied SWCNT concentrations were shown to decrease the extensibility and strength of electrospun scaffolds, so SWCNT use was considered unsuitable for this technique. The 0.01% mass fraction of SWCNT in PCL films increased the polymer strength, while fractions of 0.03% and more significantly decreased the polymer strength and extensibility compared to the undoped polymer. The PHCH polymeric films showed a similar behavior with an extremum at 0.02% concentration for strength at break.