Loose threads may break, causing you to have broken tack, or they may snag on something, which could scare the pony.
Cracked leather is more likely to break, especially if it is under pressure (such as stirrup leathers or billets).
These are what the stirrup leathers attach. Some stirrup bars can close, which is designed to prevent a stirrup leather from falling off. However, it should be left open so that the stirrup leather can come off if you fall and get your food stuck in the stirrup.
Stretched and Worn Elastic
Many girths have elastic on them. If the elastic gets to stretched and worn, it may snap, which can startle the pony and cause the saddle to fall off.
All halters should be breakaway. Breakaway halters are all leather, nylon with a breakaway tab, or nylon with a breakaway crown piece. Halters must be breakaway so that a pony can escape in an emergency without hurting itself.