Why do insurance companies often refuse to make reasonable settlement offers until a lawsuit is filed, and how does being an "eggshell plaintiff" affect a claim?

I'm in Texas and, unfortunately, I've been involved in three separate car accidents within about a year and a half because of other drivers.

First collision: I was rear-ended. The at-fault driver's insurance company refused to make a reasonable settlement offer. After my attorneys filed a lawsuit, they ended up paying their full $30,000 policy limits.

Second collision: We were in dual right-turn lanes when the other driver came into my lane and hit me, causing her vehicle to spin out. The police cited her, and there was an independent witness. State Farm initially refused to settle and eventually increased its offer to only $16,000 even though the policy limits are $100,000. My attorneys have now filed a lawsuit, and the case is still pending.

Third collision: The other driver illegally tried to pass me in a one-way lane and struck the left side of my vehicle, causing her car to spin out. The at-fault driver's insurer is GEICO. They also declined to make a meaningful settlement offer early on. The claim is still under two years old, but my attorneys will most likely file a lawsuit if it isn't resolved.

For all three accidents, my medical treatment has been well documented, and I have a police report for each collision. Since I've been injured multiple times in a relatively short period, I assume the insurance companies will argue that my injuries are related to prior accidents. I understand the concept of an "eggshell plaintiff," but I'm not sure how that applies in practice.

My questions are:

- Do insurance companies generally assume many people won't hire an attorney or will eventually give up, which is why they make such low offers before a lawsuit is filed?

- If I would be considered an "eggshell plaintiff" because of my prior injuries, does that typically hurt the value of my claim or settlement, or is the at-fault driver still responsible for the injuries they caused or aggravated?

Location: Texas