Someone buying a player with not enough money or space in team.

[GabeTC]

How can someone win a player over me, when it says they don't have enough money for what the bid was up to, and they have no space in their team?

Yes they can - if I have 1 million in cash and I bid for two players that start at 900,000 bid I can buy both of them - the result will be negative balance in my club and I have to pay it back or I can not do anything else that cost cash until the balance is paid. IE - I can't build my stadium, I can't bid for players and so on...
The other part of the question is also yes, but in that case the Manager that buys the player and has no room for it in the team will have to sell or release a player to open a spot for the new buy. Otherwise the new buy will not be in his team... you can have up to 2 extra players - (24 total) but 2 of them will not be signed in your team.