Full time recruitment advice with lack of prestigious internship

How screwed am I if I didnt intern at any prestigious big financial firms during the three of my summer during college (freshman year i interned at a startup, second year a policy think tank, third year a private wealth management office - just small private company nothing fancy). i have decent finance related ECs, very good grades (3.8+), some relevant finance classes, and wants to look for full time position, probably in investment management/ wealth-asset management, private banking, client relationship management, etc at big firms like any of the large buy side (blackrock macquarie etc), BB or boutique. If u have any advice or experiences please share! thanks