Poly(butylene terephthalate) (PBT) is difficult to foam due to its unfavorable rheological behavior (low melt strength, no strain hardening). In particular, a high expansion and a homogeneous cell morphology are difficult to achieve. This can be altered successfully by addition of multifunctional chain extenders. Chain extenders cause nondefined and rarely understood changes in the polymer architecture usually described as branching. In this contribution, the synthesis of two series of PBT copolyesters with defined short-chain branched units is presented. Dilinoleic derivatives with linear C9 and C7 alkyl side chains are employed to reflect short-chain branches and are incorporated into PBT in various molar ratios. Characterization by NMR spectroscopy and size exclusion chromatography demonstrates the random chain structure and high molar masses of the terpolyesters. Incorporation of dilinoleic derivatives results in the reduction of PBT crystallinity, decreased glass transition temperatures, and altered rheological behavior, in particular of extensional rheology characterized by strain hardening. The comparison to control copolyesters without branches proves that strain hardening is caused by the branches. A higher concentration of branches induces stronger strain hardening, resulting in successful foaming. It is demonstrated that the new terpolyesters have properties comparable with PBT treated with chain extenders.