Is there a limit to the total number of Portal Keys you can keep in your inventory?

Questions about portal key possesion in Ingress:

1. Is there a limit to the total number of Portal Keys you can keep in your inventory?
2. If yes, what happens when you key limit is reached and you got a new key via hacking?
3. Is it "legal" to hold several keys to the same portal?

There were rumors that 35 was the limit, according to this article 35 is the maximum number of portal keys including duplicates a player can hold. However the recent update confirmed that this was rather an app bug.

As of version 1.12.5 of the Ingress Scanner app, we can answer these three questions as follows:

1. There is a limit of 2000 inventory items; this would limit your number of portal keys along with other items.
2. When you're at this limit, or if you upgraded while above this limit, your ability to hack is blocked, meaning presumably, that doesn't happen.
3. It is very clearly possible to hold duplicates of keys, they are shown in your inventory with the title addendum "(n)" where n is the number of copies you own. When you scroll through the keys to make a link, each copy is it's own item card shown grouped with its like portal keys.

It's further worth pointing out that now when you hack a portal it is no longer necessary to not be holding that portal's key in order to receive a key, or a copy of the key for your inventory. E.G. without managing your inventory in any special way, you should slowly find yourself acquiring 2 or more keys for the same portal.

This version's server side changes have also made it far less likely that you'll get a key at all; I had to hack a portal I live almost on top of 20 times over two days to get a single key for it. I have yet to get the keys for 4 portals around me that were added days ago after I submitted them.

I have a friend who has a portal within the distance of sitting at his desk. He has 39 keys of just that portal. Needless to say I don't think there's a limit

Right now there is no limit on either number of different portal keys (tested up to 100+) or on the number of copies of the same key (tested up to 4) you can hold.

That is not true, you can have several keys to the same portal, the trick is to drop the existing key before you hack, gain the new one and then pick the dropped one.

-Repost of BanditoBunny's answer in a comment so we can upvote the correct answer

You can only have one portal key per specific portal, but you can have many different portal keys. At this time, it is not known if there is a hard limit of how many different portal keys a player can have. For more information:

Portals & Linking