An antipode is the opposite of a certain point on a globe, such that if you tunneled straight down into the Earth from where you're sitting the point at which you emerged would be the anitpode. In this scenario you could start in country X, drill straight through the Earth, and come out into country X again.
Sure! There are a couple of examples:
The Dutch possessed Palau Buton in Indonesia simultaneously with Suriname;
The French possessed Mauritania along with New Caledonia;
The British had Gibraltar and New Zealand;
The Portuguese might have controlled the Maluku Islands in Indonesia while administering Northern Brazil? (I don't know the time-frame on when they would have reached Northern Brazil, but it's a possibility.) Anyways, even without the Portuguese, it's abundantly clear that a number of colonial regimes controlled antipodal territory.
Source: I used this tool to find antipodes.
While it may be that I haven't been creative enough in where I'm looking, the scenario which you describe seems to be rather rare. Here are the examples I could come up with:
Makassar on the island of Sulawesi in Indonesia is antipodal to the Parque Nacional de Monte Roraima in Brazil. Portugal controlled both during the 17th Century.
However, whatever claim Portugal might have exerted over that particular far northwestern corner of Brazil was likely just that: a claim. I don't believe their actual administration extended that far beyond the coast or up the Amazon.
I learned about Makassar in Portuguese Trade in Asia Under the Habsburgs, 1580–1640 by James C. Boyajian.
The island of Luzon in the Philippines is antipodal to parts of Bolivia. The Philippines were first claimed for Spain by Ferdinand Magellan in 1521 and were then governed in various forms before being lost to the United States in 1898. Spain also controlled Bolivia as the Viceroyalty of Peru and then the Viceroyalty of the Rio de la Plata for a significant portion of the same period.
Gibraltar and New Zealand are nearly antipodal -- they miss by only a mile or so. The British Empire acquired Gibraltar after the Treaty of Utrecht in 1713 and has held it to this day, while New Zealand was at first administered as a part of the colony of New South Wales before becoming a separate colony in 1841 and later a sovereign nation.
It also looks like parts of the Cook Islands are antipodal with Egypt. The Cook Islands became a British protectorate in 1888 at which point Egypt was also under British protection.
Pretty fun map to poke around on if you're looking for antipodes: http://www.findlatitudeandlongitude.com/antipode-map/
It turns out most of this world is ocean.
The Kerguelen Islands and northern Montana, USA/southern Alberta, Canada, are antipodal. Both were French territory from 1772 to 1803 (discovery of the Kerguelens until the Louisiana Purchase).