Not as long ago as I would have thought. Per Wikipedia:

In the United States, Western Union announced the discontinuation of all of its telegram services effective from 31 January 2006.[43] Only 20,000 telegrams were sent in 2005, compared with 20 million in 1929. According to Western Union, which still offers money transfer services, its last telegram was sent Friday, 27 January 2006.[44] The company stated that this was its "final transition from a communications company to a financial services company."[45] Western Union's telegram service was acquired by iTelegram, an independent company.[citation needed] Telegrams, flowergrams, and candygrams are also offered by independent companies such as American Telegram.

I can still remember Mailgrams in the 1970's. The Post Office delivered the final message by regular carrier.

The last telegram I sent or received was when I was contacting potential employers when I graduated from college 50 years ago.