It keeps saying I don't have space, but I ugraded the spaces in my inventory. Why can't I pick things up?

You can only have a maximum of 999 of each item. So even if you have free space in your warehouse, if you have 999 Wood, you won't be able to pick any more Wood up. 

It won't let me pick up new items that I don't have :/

Go to your warehouse. 

At the top of your page it will say how many spaces you have in your inventory, and how many you have left (in the form of a fraction). 

My guess is that you inventory has filled up again. Please provide us with more information. 

Item Limit: 37/70

Interesting.... What item were you trying to pick up? Does it still happen? Are you sure the item did not exceed the 999 limit? Is it possible you can screenshot this? What about other pets picking up items, or buying from a store?

If it is not any of these things, it is possible that it could be a bug. In that case it should be posted in bugs.

I'll try to pick up a "Tapir" when exploring, which I have none of. And it says that I don't have any slots in my inventory.

I can buy things from the store thoguh, no problem.

http://i.imgur.com/6FtcPlU.png

^^Screenshot

Tapir isn't a new item for your warehouse. It's a a food resource that automatically converts to Animal Fats. Most likely your limit for this food type was reached when you tried to pick it up.

Ooooo xD Okay. That makes sense now! Thank you so much!