For example, let's say a teacher held $10,000 in the account and another $10,000 in shares of other companies (LEH acting as a broker).
Was the teacher able to get her money back after the Lehman Brothers collapse?
They would have gotten their money back.
Legally, client assets had to be segregated from the broker's own accounts. This isn't always strictly followed - MF Global notoriously dipped into customer accounts to cover trading losses before going bankrupt in 2011. In the case of Lehman Brothers, however, the Securities Investor Protection Corp (which insures against customer money going missing) announced on the day of the bank's bankruptcy filing that all of Lehman's client assets were fully accounted for.
“SPIC did not take any action because it wasn’t necessary. All customer property is there,” said Stephen Harbeck, the president of the investor protection group.
Younglai, Rachelle. (2008, September 16). SIPC says Lehman brokerage accounts intact. Reuters. https://www.reuters.com/article/us-lehman-brokerageaccounts-idUSN155295120080915
It should be noted that it was the parent holding business that declared bankruptcy, not the brokerage, which continued to function. As a matter of fact, Barclays bought Lehman's brokerage business by 19 September, with which about $92 billion of client assets (representing some 110,000 retail accounts) were transferred. The hypothetical teacher's account would have been included in this.
Institutional accounts apparently took some more legal wrangling, but five years after the bankruptcy, trustees announced that brokerage customers were getting repaid in full.
The trustee liquidating Lehman Brothers’collapsed brokerage will begin a round of distributions on Friday that should result in institutional customers getting all their money back, trustee James Giddens said.
Brown, NIck. (2013, June 7) Full payout to Lehman customers begins Friday: trustee. Reuters. https://www.reuters.com/article/us-lehman-distributions/full-payout-to-lehman-customers-begins-friday-trustee-idUSBRE9560R620130607
Ultimately, Lehman Brothers customers appears to have got all their money back. According to a press release by the SIPC,
In total, customers have received more than $106 billion, fully satisfying the 111,000 customer claims. Secured, priority, and administrative creditors have also received 100 percent distributions.
"SIPC Commends Lehman Trustee on Plans to Distribute Additional $1.9 Billion to Unsecured General Creditors" https://www.sipc.org/news-and-media/news-releases/20150713