To /u/RobBobGlove 's question first - our modern perceptions of "fat" are grossly (hah) distorted. A degree of "fatness" was indeed a sign of wealth and luxury, but the art-historical record suggests that "fat" would look more like this statue, supposedly of Charlemagne, which we might now call "slightly pudgy." Charlemagne, if the statue is an accurate representation of his body, would not have been particularly hindered by his weight. So Sam Tarly would indeed be an anomaly, and considered morbidly obese. And I can't help but noting that from a practical perspective, the relative cost of Sam keeping his weight would have been prohibitive; he should have shrunk down to a regular weight after joining the NW.

The most common mention of weight I see in the middle ages has to do with a person's holiness. The thinness of a monk denoted his austerity and dedication to his vows. Bernard of Clairvaux is described as silencing heretical critics by revealing his emaciated neck. Conversely, charges of fatness, and thus gluttony, were frequently leveled against those thought to have deviated from proper praxis. But I don't know of any examples where lay fatness was criticized by another layman.

It is a good general assumption that the nobility ate better than most, and this might make them appear to be "fat" (cf. Charlemagne) in comparison.

You might take a look at N. Koepke and J. Baten, “The Biological Standard of Living in Europe during the Last Two Millennia,” European Review of Economic History 9, no. 1 (2005): 61–95; S. Mays, “The Osteology of Monasticism in Medieval England,” in Social Archaeology of Funerary Remains, ed. Rebecca Gowland and Christopher Knüsel (Oxford: Oxbow Books, 2006).

I have to disagree with my colleague /u/SheepExplosion on this one. Game of Thrones is in many ways more parallel to the later Middle Ages than the earlier, though thus far lacking the defining technological advances (gunpowder and print) and having things like GIANT FLYING DRAGONS. Over the course of the later Latin M.A., from the 12th century onward, we can see develop through literature a growing value on an aesthetic that Vigarello calls "power and lightness." This will culminate in the first major movement of representational art in the 15th century with the reification of a strong-but-slim male figure as superior--and in the criticism even of kings for being overweight. By the time of the Renaissance, fatness is associated with sloth, gluttony, and impotence.

In the eleventh century, chroniclers do sometimes mention the excessive weight of their subjects (nobles all, sometimes royals). Fatness matters when it becomes a burden in life. William the Conqueror, whom the monk of Caen describes as great in size in every dimension (so big he couldn't be stuffed into his sarcophagus), according to William of Malmesbury died when his house bounced his excessive paunch against the pommel of his saddle and his intestines ruptured. Orderic Vitalis, who offers what is generally considered the best description of William's death, does not moralize the death or the fatness. He portrays William as a clear-headed, intelligent, wise-ruling king right up until the end of his death from infection.

Interesting, William of Malmesbury seems to have had an eye for the corpulent. It's his testimony that tells us King Philip of France used the excessive obesity of his wife as a reason to repudiate her (a cause of infertility); meanwhile, Philip was increasingly too fat to ride normal horses.

The seeds are planted: the idea that a man can become too fat for service in battle. In the twelfth- and thirteenth-century romances, we occasionally get a description of male characters beyond "beautiful" (which, in the Middle Ages, basically means young). Tristan of the Arthurian court is described as broad-shouldered and slender-hipped (swimmer's body, amiright, ladies?)--the physique of a mounted knight, who must be light and strong, agile and steadfast. Likewise, Reynault in the Belle Erembourc has broad shoulders and a flat stomach.

From the twelfth century onward, as well, the Church's ongoing efforts to Christianize the laity through, above all, moral instruction via the Seven Deadly Sins and the sacrament of confession drew increasing attention to the cause of fatness: excessive consumption, the sin of gluttony. We know from medieval texts that medieval people were highly aware of the link between overeating and overweightness, and the difference between muscle and fat. (That probably doesn't need to be said, but just in case.) The iconographic symbol of gluttony is the fat man.

The clergy's efforts, one assumes, contributed to the moralization of fat. Fatness is less the result of wealth and a life of deserved excess, and more the result of spiritual failing. The moral thing to do is not to eat in overabundance; it is to maintain sobriety in the face of possibly abundance.

By the fifteenth century, the game is on. In 1457, the Duke of Milan consulted a doctor for advice because his son had grown too fat. Chronicler Philippe de Commines will describe the late King of England thus:

He indulged himself in [his pleasures] more recklessly than before. From this time he feared nobody, became very fat and full. At a young age his excesses overtook him and he died of apoplexy.

A final piece of really interesting evidence comes from Gaston Febus, Count of Foix's, Livre de chasse. One 15C manuscript of it illustrates a feast scene. The overweight servants and valets tear into their food. The high nobility, slender and strong, eat calmly and regally. Male fatness isn't just a marker of morality, it's a marker of manners, power and class.

Throughout the later Middle Ages, people were well aware that excessive fatness could prove a burden in battle and could be unhealthy. But it was a process of development that moved from that recognition to an association of fat with moral failure, especially on the part of lords who had to rule.

I think that it would probably be fair to assume that the average member of the nobility or royalty in the early and central Middle Ages was not likely to have been overweight, and that the striking examples of a few fat individuals were worthy of note precisely because they were the exception to the rule.

As recorded for tenth-century Iberia in chronicles from both the Latin north and Arabic south, a Leonese king known as Sancho the Fat (el Craso) was said to have been unable to rule because of his extreme weight, too fat even to sit a horse. After being deposed by his rebellious noblemen, he looked for help to al-Andalus, and in Cordoba he received medical treatment from the court scholar-physician Hasdai ibn Shaprut. Perhaps more significantly, following his successful weight loss, Sancho also received military support from the caliph ‘Abd al-Raḥmān III, which enabled him to take back his throne.

The following summary of the situation, including the key diplomatic role played by Queen Toda of Navarra, who was the grandmother of Sancho and the aunt of ‘Abd al-Raḥmān III, is from Glaire Anderson, "Sign of the Cross: Contexts for the Ivory Cross of San Millán de la Cogolla," Journal of Medieval Iberian Studies 6:1, 15-41, DOI: 10.1080/17546559.2014.889321

"One of the most intriguing and oft-repeated stories told about Andalusi and northern Iberian interchange is that of Queen Tota’s appeal to the caliph to aid her grandson Sancho I 'the Fat' (r. 955–57 and 960–66), king of León. Sancho had been dethroned in 957, his claim to the crown of León threatened in part by his obesity. Despite having dealt ‘Abd al-Raḥmān III a devastating military defeat at Simancas (Arabic Shant Mankash) or Alhándega (Arabic al-Khandaq) just six years after his legitimization of her son’s right to the throne of Navarre (hence, presumably, Ibn Ḥayyān’s characterization of her as cunning or deceptive), it was to Córdoba that Tota turned for help in aiding her grandson to regain the throne of León. Her decision is suggestive of Córdoba’s continuing political and ideological strength, despite the serious military defeat, and it gestures to the way alliances across religious borders could be reasserted when it was advantageous for one or both parties to do so. Having the rulers of both Navarre and León in the role of supplicants, following on the heels of the Simancas debacle would certainly have been advantageous to ‘Abd al-Raḥmān III. Once more ‘Abd al-Raḥmān agreed to assist Tota in maintaining control of a northern crown, providing Sancho with the expertise of his own doctor, the celebrated Jewish physician Hasday b. Shaprut. The treatment successful, Sancho regained the Leonese throne, which he ruled from 960 until his death in 966."

For the Christian perspective from the Chronicle of Sampiro: --Historia Silense, eds. Justo Pérez de Urbel and Atilano González Ruiz-Zorrilla, Madrid, 1959. --Justiniano Pérez de Urbel, Sampiro, su crónica y la monarquía leonesa en el siglo X, Madrid, 1952.

For the Islamic side, which comes from Al-Maqqarī: --Al-Maqqarī, Aḥmad ibn Muḥammad. Analectes sur l’histoire et la littérature des arabes d’Espagne, ed. Reinhart Pieter Anne Dozy, 2 vols., Amsterdam, 1967. --Évariste Lévi-Provençal, Histoire de l’Espagne musulmane, 3 vols. Paris, 1950.

bonus: what was fat in those days?

I'm on mobile and having trouble remembering the rules, so if I mess up a top level comment, I apologize, but I'd like to add a follow up question.

In Game of Thrones, Sam is not a noble. We meet him after he's been stripped of lands and titles and sent to the wall (that's not a spoiler, we meet him very early in the first season).

Would a non-noble be mocked for his weight? Much of that first season's interaction among young men at the wall is creating a pecking order and establishing in-groups and out-groups. Would young men have used weight as a way to do so?