It depends on the army I suppose, also what you class as a "Gentleman".

If you define that as someone with an education, but little money or connections, then there are some examples in the British army (the 2nd bat. 95th had maybe 1 or 2 examples in a battalion of over 900 men- see Mark Urban's book Rifles). It is worth remembering that while there are examples, as a whole this was only a minuscule percentage of the British army. Most men from educated backgrounds in the British Army came from the upper or emerging middle class and so would have purchased a commission (albeit in unfashionable regiments for the latter, as these were often much cheaper).

This was because of attitudes towards the army this time. As a rule the army & it's soldiers were much less popular in society than the Navy, especially following Trafalgar. Generally speaking, rank and file British soldiers were not looked on favorably in society partly due to the fact that enlistment was often a punishment for petty crime or a way out of poverty. However, being an officer was seen as fairly respectable career, so if you could afford it you did it, even if it meant scraping together money for an unfashionable regiment.

For an aspirational middle class intent on proving their position in society, the done thing would be to buy a commission, otherwise it would be seen as a step back rather than a step forward socially. For the highest levels of British society, joining as a ranker would have caused quite the scandal.

As for the armies of less industrializing countries at the time (such as Austria, Russia and Prussia), I imagine that they would have been a bit more rigid in terms of their social makeup, due to the absence of the minor social mobility that came with the Industrial Revolution in Britain (e.g. a large, emerging and aspirational middle class). Anyone from any form of more privileged background would have had the means or the social circle to buy or arrange a commission - it was just the done thing.

The French on the other hand introduced conscription or the 'levee en masse' at the start of this period - so the possibility of educated men in the ranks therefore is much higher. However, often educated or wealthy men could go straight to the rank of officer. Phillipe de Segur (later known as Phillipe Paul Comte de Segur, a minor noble whose family had survived the purges of the revolution), recalls in his memoirs how he was made lieutenant when he enlisted in 1800 because he not only convinced some men to join with him but also had a formal education. While I'm sure gentlemen rankers were much more common in the French army than their British or Russian counterparts, I wouldn't be able to say with confidence that there were a large number of them.

Hope that helps - I recommend Richard Holmes' Redcoat as a book you may want to read if you are interested in the British army in this period