We only know of a handful of direct interactions between Chinese and Romans, as they traded through numerous intermediaries in Central Asia or the Indian Ocean and not directly (there was significantly more contact between the Roman Empire and the west coast of India, for example). We should think of the Roman and Chinese empires as the eastern and western poles of a broad web of trade rather than the two ends of a straight line.

To answer your question more directly, trade between East Asia and Western Europe did not stop after the fall of the Western Roman Empire, but there was a decline in volume due to upheavals in the West (fragmentation, population decline, a shift towards a manorial economy), the Eastern Empire (the plague of Justinian, the long war with Khusrau II, the Arab conquests), and in China (dramatic population decline in the late Han and Wei, followed by the political disorder of the Eastern Jin). Eurasian trade fluctuated along with the demographic and economic fortunes of each of its participants but at no point did it cease completely.