James, I have done this several times flying to/from Europe, on the same day. Yes, it is risky - more risky depending on time between flights. I have done it both to save money and for convenience. E.g., years ago, when I started a trip to Poland, I got a direct flight into Amsterdam (pretty sweet to get a direct flight to Europe), and a few hours later, I caught a direct flight from there to Gdansk, my starting point. This was the most optimal way to get to Gdansk. Flying any other way would have required at least one more stop and hours longer to get to Gdansk. Of course, had I arrived late in Amsterdam and missed my connection, it would have been much worse.
This kind of question really has two dimensions and I think you should separate them: 1) what are the logistics of traveling on two separate tickets, how do I estimate the risk, and how bad is my backup plan? (Yes - you have to have a backup plan and be willing to accept it if you need it.)
And 2) what is my personal tolerance of risk? This one is important when reading people's responses...because THEIR tolerance of risk may be different than YOURS! Just because someone else would NEVER fly on two tickets on the same day doesn't mean YOU shouldn't! Some people are extremely risk-averse, and that will color their answers.
In my case, my backup plan in case I miss a connection (and lose the value of my second ticket) is usually to buy another plane ticket the same day...or at worst, not get to my destination the same day, just spend the night in the connecting city and get there the next day. Usually I've got a few nights in the first city, so arriving a half a day late would be unfortunate but not a disaster. If I was catching a cruise ship the next day that would leave without me, that would be a disaster, right?
Sometimes the backup plan is not bad. Maybe there are additional direct flights the same day after the original one you are booking. It might cost a few hundred Euros. If it would cost a thousand Euros, that might be a lot different.
But it's important to consider what affects the risk. Is the connecting airport one with a bad reputation? Hard to connect in? Will I have checked bags? Can I move quickly between connections? (I travel with carry-on bags only and can more pretty fast with them in an airport. Until last year's summer meltdown at Schiphol airport in Amsterdam, it had a reputation for smooth connections with minimal delays, usually without any shuttle buses, etc., so I felt pretty good about doing this type of connection there.)
Sometimes trip delay insurance may cover the costs of a delay caused by a delayed flight. Check with any insurance provider that might cover it.