Lots of nations celebrate the day then gained their freedom from some foreign empire - usually either Britain, Spain, Russia, or France.
Mexico - Grito de Dolores
Mexico's real independence day is September 16, 1810. On that day a priest, Miguel Hidalgo y Costilla, in the small town of Dolores in central Mexico addressed a gathering outside his church where he urged the townsfolk to rise up in revolution and kill their Spanish rulers (the "Cry of Dolores").
He marched his people across Mexico gathering more followers as he went, growing from 600 to over 80,000 rebels. They killing Spaniards wherever they found them. In 1811 his rag-tag force of peasants was met and routed by a few thousand trained Spanish soldiers. Hidalgo was captured and executed by firing squad. The rebellion Hidalgo had started could not be quenched and Mexico was declared independent of Spain in 1821.
Israel - Yom Ha'atzmaut
Vietnam - Ngày Quốc Khánh
Later that month, Nationalist China invaded northern Vietnam to hold that part of the country until the French could re-exert their colonial power. In the south, British forces invaded and freed captured Japanese soldiers to help them subdue the Vietnamese. This began the partition of the country that would be the crux of the prolonged war.
By 1950, France had reinstalled the puppet emperor that Japan had used in Saigon while now Communist China supported Ho Chi Minh in Hanoi as the true leader of the country. Eventually, the United States replaced France as the colonial power controlling a puppet government. The final defeat of the US in 1975 is only considered the culmination of the independence struggle begun 30 years before.
Latvia - Two Independence Days
World War I and the Communist revolution shattered the Russian Empire. Latvia and the other Baltic countries had been part of Russia since the early 18th century. In March, 1918 the new Communist government of Russia was desperate to end the war with Germany, The Treaty of Brest-Litovsk gave Latvia and the other Baltic countries to Germany. Germany's surrender several months later allow the Baltic states to declare their total independence.
Fast forward twenty years. The Nazi-Soviet Pact of 1939 divvied up eastern Europe between them. The Soviets got the independent Baltic states and shortly afterwards Russia sent in unwelcomed troops to "protect" Latvia. They forced Latvia to hold an election where they were given the choice of joining the Soviet Union peacefully or joining the Soviet Union violently. World War II did not change that annexation.
Fast forward a half century to 1990. Like the Russian Empire before it, the Soviet Union is collapsing. Latvia declares the Soviet annexation of their country null and void and that they are again an independent state. May 4 is now celebrated as Restoration of Independence Day in Latvia.