The short answer is 'yes'.
"I'm considering buying a new car" is something you'll hear said quite often, and it's perfectly grammatically (see what I did there?) correct.
While "I'm planning going on a diet" is grammatically correct (using an '-ing' verb can work with any otherverb following it is correct), it is not considered good use of English, just as "always isn't" instead of "never" is bad use of English if you can use something else instead, although in this case I believe it is nothing more than the sound which sounds wrong.
In the former example you gave, the two '-ing' words have a different syllable count, while the latter example both are single-syllable words. This is a simple case of the former example sounding fine, whereas the latter example just sounds as if you are repeating yourself. Thus, it is considered bad use of English.
This isn't unique to 'ing' words or nouns, though. You may have noticed I used the words 'perfectly' and 'grammatically' together. It doesn't sound proper but is nonetheless legal. I simply couldn't escape using it in this instance.
As alluded to by others here, the latter example, "I'm planning going on a diet" (which should actually be "I'm planning on going on a diet"), will almost without exception be phrased either "I'm going on a (new) diet" or "I'm going to go on a diet" (used when emphasising intent more than what you are intending to actually do).