Simply put, the answer is yes-just not in the way you think. The fat Buddhas you see in restaurants are importations from Chinese folklore, sometimes by way of the Japanese. They depict the Chinese monk Pu-Tai, or Hotei in Japan. He was a monk who wandered the Chinese countryside, known for his generosity and vision; he may or may not be a conflation with the Chinese god of prosperity, luck, and happiness.
The size of the fat Buddha's belly represents his great spirit of generosity and perhaps compassion; his belly holds "many souls." Due to its connection with physical sustenance and pregnancy, in many religions, the belly is an important center of spiritual power. The belly is often seen as the truest center of a person, and rubbing the Buddha's belly can often bring luck- even fertility.
However, this Buddha has no physical connection with Gautama or Siddhartha Buddha, the historical Buddha and founder of Buddhism. They were different people. The common misconception that the Buddha became fat and prosperous after renouncing the ascetic paths of fasting and self-denial seems unlikely. The historical Buddha insisted on a middle path in this regard, renouncing both the Hindu asceticism and the hedonism and self-indulgence of the nobility. He had lived both of these ways, and rejected them after reaching enlightenment, preaching instead practical moderation and restraint in all things.
Beyond this level of description, however, the issues of identity in Buddhist thinking become more complex. First, there is the Buddhist teaching that all are one- novices in Buddhist monasteries are often pointed to statues of the Buddha (fat and skinny and muscled and varied in many other ways) and told "that's you." So in this way, regardless of their physical and historical distinctions, the fat and skinny manifestations of Buddha do represent the same spirit, if not the same person: one of the many distinctions not as clear in Eastern thought as in the West.
Furthermore, some sects of Buddhism believe in past, present and future Buddhas- with the future Buddha being the more rotund. Not only are these one in the sense that all people are one, but also in the sense that Buddha can have more than one manifestation; they are the same Buddha, the same Enlightened One. Given that anyone can achieve enlightenment, the potential number of Buddhas is infinite.
Or one, depending on your perspective.