On several of my aloe vera plants, the leaves suddenly became kinked. The leaf would get very thin in a spot and bend over. It starts near the tip and is most noticeable on older leaves. What is causing this? It happens inside more often than out, and it isn't the cat - he isn't allowed in the house.
From comment: It happens indoors more often than out. I do not think it is my cat. He is not allowed in the house.
From here: The soil is good, dark, crumbly garden soil that grows plants very well.
If the Aloe Vera in your question has the same potting medium as quoted above, I believe over watering or the high moisture retention properties of the "potting medium" is the most likely cause of the tips curling up.
The culprit may be in the background of the photo. I've got a cat that chewed on mine occasionally before I got him a pot full of grass to chew on instead.