Short answer
Yes. The Prophet Muhammad (ﷺ) named it as a sign of the Last Day. He said that barefoot, destitute herdsmen would compete with one another in raising tall buildings. The herders his hearers knew were the Bedouin, the desert Arabs who herd animals. They were among the poorest people of the known world and lived in tents. The great builders of that age were the Romans and the Persians, not desert herders. For thirteen centuries the herders of the Arabian desert stayed poor. Then, from the 1930s, oil was found in the Gulf. Today the tallest building on earth stands in Dubai, and towers rise across the Arabian peninsula. Their cities still compete to build higher. The saying is in Sahih al-Bukhari and Sahih Muslim, the two most trusted books of hadith. Islam's answer is that this is one sign that he spoke by revelation from Allah (﷾).
Full explanation
What the Prophet (ﷺ) said
The saying comes from the hadith of Gabriel, one of the best known reports in Islam. A hadith is a report of what the Prophet (ﷺ) said or did. In this one, the angel came in the form of a man and questioned the Prophet (ﷺ) before his Companions. Asked when the Hour (the Last Day) would come, the Prophet (ﷺ) said the one asked knew no more about it than the one asking. Then he gave its signs. Umar, a Companion, reports that the builders would be barefoot, poorly clothed, destitute shepherds, competing with one another in raising tall buildings. Abu Hurayra, another Companion, tells the same event in Sahih al-Bukhari.
Umar ibn al-Khattab reports that a stranger in very white clothes came and questioned the Prophet (ﷺ) about Islam, faith and excellence in worship, and then about the Hour. The Prophet said the one asked knew no more than the one asking, and gave two of its signs: a slave woman will give birth to her mistress, and you will see barefoot, poorly clothed, destitute shepherds competing with one another in raising tall buildings. Afterwards the Prophet told Umar the stranger was the angel Gabriel, who had come to teach them their religion.
Abu Hurayra reports that the angel Gabriel came to the Prophet (ﷺ) in the form of a man and asked him about faith, Islam and excellence in worship, and then when the Hour would come. The Prophet answered that the one being asked knew no more about it than the one asking. He described some of its signs, said the Hour is one of five things that no one knows except Allah, and recited Qur'an 31:34.
Who he named
A guess that cities will grow is easy to make. This saying is different. It described the most unlikely builders in the world of its day: barefoot, destitute shepherds. The herders his hearers knew were the Bedouin of the Arabian desert, who owned little beyond their animals and lived in tents. The great builders of that age were Rome and Persia. For about thirteen hundred years after the Prophet (ﷺ) spoke, the herders of the Arabian desert itself stayed poor and lived in tents. Nothing in their lives pointed to towers.
Umar ibn al-Khattab reports that a stranger in very white clothes came and questioned the Prophet (ﷺ) about Islam, faith and excellence in worship, and then about the Hour. The Prophet said the one asked knew no more than the one asking, and gave two of its signs: a slave woman will give birth to her mistress, and you will see barefoot, poorly clothed, destitute shepherds competing with one another in raising tall buildings. Afterwards the Prophet told Umar the stranger was the angel Gabriel, who had come to teach them their religion.
What happened
Oil was found in Bahrain in 1932, and in Saudi Arabia and Kuwait in 1938. Within a single lifetime the grandchildren of desert Arabs were ordering towers. The Burj Khalifa in Dubai, at 828 metres, has been the tallest building on earth since 2010. Saudi Arabia's Jeddah Tower is being built to pass a kilometre. Other towers show the cities of Arabia competing with one another to build higher. They include the Abraj Al Bait in Mecca, the Kingdom Centre in Riyadh and Al Hamra in Kuwait.
A history of the oil industry. It describes the first oil strike on the Arab side of the Gulf, in Bahrain in 1932, and the great discoveries in Kuwait and Saudi Arabia in 1938.
The Council's database, the standard record of the world's tallest buildings, lists the Burj Khalifa in Dubai at 828 metres, completed in 2010, as the tallest building in the world.
The Council's database lists the Jeddah Tower in Saudi Arabia as under construction, planned to be the first building over a kilometre tall, and lists among the completed towers of the peninsula the Makkah Royal Clock Tower (Abraj Al Bait) in Mecca, the Kingdom Centre and Al Faisaliah in Riyadh, and the Al Hamra Tower in Kuwait City.
Perspectives
The Islamic perspective
The Prophet (ﷺ) foretold something no one in seventh-century Arabia could have expected, and it came true in the twentieth century. It is one of many signs that he spoke from revelation.
In the hadith of Gabriel the Prophet (ﷺ) said that barefoot, destitute herdsmen would compete in raising tall buildings.
The Burj Khalifa in Dubai, at 828 metres, has been the tallest building in the world since 2010.
The saying described barefoot, destitute shepherds like those of the Arabian desert, not the empires of the day. That makes it specific, not a general guess.
The fulfilment of this sign is part of the evidence that the Prophet (ﷺ) was sent by Allah (﷾).