This tile end dates to the Western Han dynasty. The characters on the tile refer to the founding of the Han dynasty. This tile was made during the earlier part of the Han dynasty which is referred to as the Western Han dynasty. The Han dynasty actually ran from 206 BC to 220 AD or 427 years. That period includes the Western Han, the Xin, and the Eastern Han dynasties.
The "Hàn Bìng Tiān Xià" tile shown above carries the slogan or saying declaring that the Han were going to merge or unite the rest of the States. It may have been used like the campaign slogan "Tippecanoe and Tyler Too!" to rally enthusiasm and support, and perhaps make a prediction. Later placement of the slogan on the tile was a method to declare the Han destiny for all to see. The slogan refers to Liu Bang who started out as a minor functionary in the Qin dynasty and eventually overthrew that dynasty to become the first Han emperor. As the slogan says, he did unite all of the States under his rule to produce one of the great dynasties, but we don't know if the tile describes his fate or if one of his ministers made it up to garner support from both the military and populace. It makes a good rally cry and an even better prediction.
Even before the first Qin emperor died his kingdom was rife with rebellion and dissatisfaction. The first Qin emperor, Shi Huang Di. was in the habit of conscripting volunteers to go to the frontier to build the Great Wall, dig various canals, and construct his mausoleum, to name a few. Millions were displaced into what was slavery under the harshest conditions. While the Qin dynasty achieved great things, it did so at greater human cost.
When the first emperor died, the various States which had been coerced to follow his rule began plotting to achieve independence or supremancy for their own little corner of the world. Liu Bang became a general who successfully captured the stonghold of the old Qin dynasty and thus played a key role in the end of the struggle. Rather than the honor and reward he had been promised, he was sent off to the hinterland to govern a minor area. That didn't sit well with him, so he gathered supporters and counterattacked.
"Hàn Bìng Tiān Xià" either declared Liu Bang's intention to rule the whole of China or predicted that he would. It is most likely that one of his advisors came up with the slogan during the campaign and then, when they were successful, turned it into a prediction to make it seem that Liu Bang was destined by heaven to be emperor and to unite the States.
The Han dynasty was one of the great periods of development in China. While the Qin was the first, it was short-lived and noted for its violence and lack of successful peacetime governance. The Han dynasty did much to cement the concept of the "central kingdom" in the minds of Chinese, establish and ethic, a philosophy, and a common The economy was good, art flourished, great civil works were completed, and the first official forays to the west along what would become known as the Silk Road were made. Over 400 years of relative peace allowed the spread of culture and religious beliefs and many of the local variations in each began to smooth out. The common belief system was actively encouraged by the early Han emperors when it was decided to promote Confucian ideals as a form of religious practice. All of that comes to mind for a Chinese school kid when they hear the phrase "Han Bing Tian Xia."