Could I dye my hair again?

Okay, i've asked this question before, but wasn't too happy with the answers.

In April, I bleached and dyed chunks of my hair blue (my bangs/fringe and just random pieces)

Then in June I dyed over the blue chunks red and purple.

Now that the red and purple are faded into a faint purplish blonde color, I'd like to dye it one more time.... But I would have to bleach it. I asked my mom if I could, and she said she'd see how damaged it is after I get it cut. Yesterday, my grandma looked at my hair and asked me if I ever deep condition it because it's all frizzy, I actually never do. Could I bleach the dye my hair one more time? Because I'd really like to, before school starts.

Thank you!