Is there any truth to this?

Yes. Modern horses are the product of centuries of selective breeding, to foster strong, high-endurance and fast animals, and are aided by having high standards of veterinary care, not just in keeping them fit and healthy, but also improving fertility, allowing for faster breeding (Johnstone 42-43). In much of the ancient world, the emphasis was on 'breaking' horses rather than Mengelian-type breeding ('breaker of horses' is a common epithet in the Iliad). The later Romans did breed horses, but not as extensively as in the Medieval periods. They took precautions to maintain their health and promote growth, but health problems were common (Johnstone 44-46).

Romans certainly recognised that some horses were more equal than others, and observed differences between horses sourced from different regions. Gallic horses were regarded by Caesar as 'small and ugly'; by contrast, Parthian/Persian horses were larger and stronger (Johnstone 40-41 contains recorded observations of this sort). We should note that since horses were kept for different purposes, so a substantial disparity between a champion race horse and a 'beast of burden' should be expected.

Sources and further research:

C.J. Johnstone (2004), 'A Biometric Study of Equids in the Roman World' (PhD Thesis, Department of Archaeology, University of York). Pages 1-136 available here.

C. Jones (2012), 'Horses: archaeology, history and myth' (audio), given for the British Museum exhibition 'The horse: from Arabia to Royal Ascot'.

Horses in Late Antiquity were rather small in the Persian and Roman Empires. Here is an image of Sassanid Emperor Shapur I receiving homage from the captured Roman Emperor Valerian. The horse Shapur I is riding looks to be pony sized. This is from a rock carving located at Naqs-I Rustam, seven kilometers south of Persepolis.
http://usna.edu/Users/history/abels/hh381/ShapurI_Valerian.jpg