The radius and surface composition of an exploding massive star, as well as the explosion energy per unit mass, can be measured using early ultraviolet (UV) observations of core-collapse supernovae (CC SNe). We present the results from a simultaneous Galaxy Evolution Explorer (GALEX) and Palomar Transient Factory (PTF) search for early UV emission from SNe. We analyze five CC SNe for which we obtained near-UV (NUV) measurements before the first ground-based R-band detection. We introduce SOPRANOS, a new maximum likelihood fitting tool for models with variable temporal validity windows, and use it to fit the Sapir & Waxman shock-cooling model to the data. We report four Type II SNe with progenitor radii in the range of R
* ≈ 600–1100 R
⊙ and a shock velocity parameter in the range of v
s* ≈ 2700–6000 km s−1 (E/M ≈ 2–8 × 1050 erg/M
⊙) and one Type IIb SN with R
* ≈ 210 R
⊙ and v
s* ≈ 11,000 km s−1 (E/M ≈ 1.8 × 1051 erg/M
⊙). Our pilot GALEX/PTF project thus suggests that a dedicated, systematic SN survey in the NUV band, such as the wide-field UV explorer ULTRASAT mission, is a compelling method to study the properties of SN progenitors and SN energetics.