No.
When adding the smallest 2 digit number (= 10) to the smallest 2 digit number (again, 10) the result is a 2 digit number (10 + 10 = 20).
When adding the largest 2 digit number (= 99) to the largest 2 digit number (again, 99) the result is a 3 digit number (99 + 99 = 198).
As you can see, you'll either get a 2 or 3 digit number, but never a 4 digit number.
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Well, friend, when you add two two-digit numbers together, the sum will always be a two-digit number or a three-digit number, not a four-digit number. It's like mixing different colors on your palette - you'll get a beautiful new shade, but it won't be something completely different. Just keep adding those numbers together and creating something wonderful!
Add the two greatest possible four digit numbers. 9999 + 9999
Four. (100 x 10 = 1,000)
The number of digits increases - without limit.
If you realize that a 4 digit palindrome is basically a two digit number followed by it's own reverse, you can re-phrase this question as 'how many two digit numbers are there?' Which would then mean that (if you're considering zeros as digits without a non-zero leading term) there would be a 4 digit palindrome 0000 for the two digit 00; a four digit 0110 for the two digit 01; 0220, 02; etc etc. So the number of four digit palindromes would be the same as the quantity of whole numbers from 00 to 99. So I believe the total answer to your question would be 100.