Yes, you can have an overflow and a carry flag in the same operation. This is because an overflow flag is hooked up to an XOR gate, where the inputs are the carries from the two bits on the left. The carry flag gets triggered if there is a carry on the left most bit (the signed bit for signed numbers). So if you have a carry on the left most bit, but not on the one second from the left, you have an overflow and a carry. If you have a carry on the bit second from the left and not on the left most bit, you have only an overflow.
Here's a picture of an adder/subtractor where the V is the overflow, and the C is the carry. Each bit of the sum is calculated using a full adder, and each bit may have a carry. V is an XOR gate connected to the two left most carries.
What the overflow represents is when an operation causes the sum to fall outside the maximum or minimum bounds. For example, with an 8 bit signed operation, the maximum number of the sum is 255. while the minimum is -256.
So if you try to do 255 + 1 (01111111 + 00000001) you get an overflow. Notice how there was a carry on the second bit from the left, but not the left most. The sum comes out as -256, which is wrong.
An example where you get both an overflow and a carry would be -64 + (-256) (11000000 + 10000000). There is a carry on only the left most bit. The sum comes out as 64, which is wrong.
An example where you get only a carry would be -64 + 64 (11000000 + 01000000). There is a carry on both the left most and second from the left. The sum comes out as zero, which is right.