Hello kaffwyn! No. But sort-of. There is one interesting cases I can think of that sort-of fit the bill of “negative energy” but really doesn’t!
The first is antimatter. Antimatter was actually predicted before it was discovered. It just sort of naturally appeared out of the equations describing relativistic electrons (i.e. combining special relativity with electrons – so electrons at really fast speeds!) first written down by a physicist named Paul Dirac. At first it looked like writing a relativistic equation for the electron indicated they could have negative energy. Solutions of the Dirac equation (the special equation describing relativistic electrons) contained negative energy quantum states!! As a result, an electron could always radiate energy and fall into a lower and lower negative energy state! However, negative states have no floor of minimum energy (like a minimum value of zero provides). So this was a big problem! Dirac was clever though. He figured out if a new particle existed, with identical properties to the electron, but with opposite charge, those negative energy states wouldn’t be a problem. Dirac called these new particles “negative energy electrons” which is a bit of a misnomer but that is certainly how they look in his equation. Turns out these particles are real – they are the antiparticle of the electron, the positron! (Note: positrons have positive energy – it is just some subtleties in the equations that make them look like negative energy electrons).
A far as we know, no real particles or physical processes have negative energy! That would be really really bad!! Great question! Keep it up!