Is there a limit on how fast my Handicap Index can go up?

Yes, once you have a Low Handicap Index: the lowest Index you've had in the 365 days before your most recent round. It's established once you have at least 20 acceptable scores (Rule 5.7).

• Soft cap: if your new Index would be more than 3.0 strokes above your Low Handicap Index, only 50% of the increase above 3.0 counts (Rule 5.8).
• Hard cap: after the soft cap, your Index can't be more than 5.0 strokes above your Low Handicap Index.
• There's no limit on how fast your Index can go down.

Other limits: the maximum Handicap Index is 54.0 (Rule 5.3). Your Index updates no later than the day after you post (Rule 5.4). A score stays in the calculation as long as it's among your 20 most recent, however old, and your Index only lapses if you stop being a member of a golf club (Rule 5.5).