If you can't name them all without checking your notes, as player or GM, that's a good sign you have too many.
A recent thread on The RPGnet asked for suggestions to add to the roster of a superhero secret agent support team, and this reminded me of something a wise man (hello Dave!) once said, about how James Bond works for MI6 but he only interacts with about three people from this organisation staffed by hundreds. They all have strongly sketched in personalities and they're memorable. Having a different briefing officer for every adventure, and a different member of Q Branch, would dilute the flavour of the setting.
Give the organisation a boss, with some interesting eccentricities and foibles, and a small number of other NPCs just as broadly drawn.
A large organisation might have specialists on call for specific weird situations. If they can be reached as soon as the PCs need them (directly or by phone or the like) this can result in the PCs using "call a specialist" as a crutch, rather than, say, going in and trying to defuse a bomb themselves with no demolitions training or tools like adventuring heroes are supposed to. So it often helps to strand the PCs away from direct contact with backup. Or if that feels too forced, just limit the availability of specialists from the get-go.
Delta Green, for example, lets PCs work as members of government agencies, but as they're working an unofficial and illegal agenda they can't call in too many favours. Angel has the organisation rules, where the GM and players decide what resources they have - and since you can limit the points they have to spend, the bigger a group is the less PCs will matter to it.
If the PCs know in advance they'll need a specialist for a mission, that can bring an NPC into the party - feel free to characterise them pretty broadly. And if you need someone to be killed to show how this week's monster works, they're always a good choice...
There's also the possibility of an NPC full-time member of the party. Avoiding the horrors of the GMPC and making ure they don't hog the spotlight, this character can be very useful. You can fill a gap in the skillset of the group, possibly taking over a necessary but unwanted role like getaway driver, and it gives you a direct voice in planning (and arguing, and making jokes) in-character, and it's someone you can hit with plot devices as much as you want.
No magic-user among the PCs? Well, the local witch might be friendly, but she can also mess up at just the wrong moment bringing in magical problems the group wouldn't otherwise suffer, has her own agenda, and would make an interesting wedge in the relationship between two PCs...