There are a few important downsides to drinking too many energy drinks. This is not to say that drinking these products is horrible for you and should never be done. For the most part they are like many things: okay in moderation, bad for you in excess.
Energy drinks contain a lot of caffeine and other stimulants. Many of the also contain a lot of sugar which can have it's own detrimental effects on the body. The caffeine isn't necessarily bad for your brain or your stomach. It can raise your heart rate and blood pressure and causing heart problems over time. While very few people can attribute their heart disease exclusively to caffeine, it is well recognized as a contributor.
The other negative effect it has is on your ability to sleep at night. Any amount of caffeine that you consume in the afternoon is likely going to decrease the amount of good sleep you get at night. The less sleep you get, the more tired you will be the next day. This may lead to more energy drinks consumed. Thus, these drinks can lead to a bad cycle of inadequate sleep and feeling more fatigued.
This is a good topic to bring up with your primary care physician
. He or she will be able to give you the best advice on this topic, with your own personal medical issues in mind. Good luck.