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The Census Bureau last week reported that ownership increased to 63.9% in the third quarter, the highest level since 2014. The rate was up from 63.7% in the second quarter and 63.5% a year earlier. It remains below the 69% clocked at the peak of the housing bubble a decade ago.

Still, the recent trend is causing analysts and investors to wonder whether the rental market’s good times are coming to an end. Investors pumped tens of billions of dollars into the sector during the recovery, building and buying apartment complexes and amassing large portfolios of single-family homes.

One early warning sign came last week, when American Homes 4 Rent, which owns more than 50,000 single-family properties across 22 states, reported disappointing revenue growth for the third quarter. Analysts will be closely watching earnings from two other big companies with similar portfolios. Invitation Homes Inc. and Starwood Waypoint Homes, which agreed in August to merge, will both report results on Wednesday.

“Up to now, there really hasn’t been a chink in the armor of rent growth,” said John Pawlowski, an analyst with Green Street Advisors.

Many analysts predict that any pain that rising homeownership causes to the rental sector will be felt by these companies first, before renters of luxury apartments in big cities, for two reasons. First, house-for-rent companies tend to own properties in more affordable, nonurban markets. Second, people living in such homes have already opted for the single-family home lifestyle and so are more likely to become a homeowner.

To be sure, multifamily investors aren’t headed for the door. The homeownership rate is still well below the historic norm of 65%, and growth could be slowed by such forces as rising interest rates and last week’s tax code overhaul proposed by House Republicans.

A spokeswoman for American Homes 4 Rent said: “We believe our country’s cumulative undersupply of housing stock, along with shifting preferences towards rentership provides a favorable landscape for single-family rentals into the future.”

Still, the rise in homeownership comes as other forces weaken the rental market, including a surge in supply from developers hoping to cash in on rising rents. In September, the seasonally adjusted rate of apartments under construction was 596,000, nearly twice the long-term average of 300,000 units, according to U.S. Census data.

Job growth also is slowing in some markets. That, coupled with new supply, boosted the national vacancy rate to 4.5% in the third quarter of this year, compared with 3.5% a year earlier, according to John Chang, head of research for real-estate services firm Marcus & Millichap. Nationally, rents were up 3.5% between the third quarters of 2016 and ’17, compared with 4.5% the previous years, he said.

In many markets, developers completing new projects are offering concessions to woo new tenants, such as one or two months free rent. Lately, the trend has spread: Landlords in some markets are reporting that owners of previously built properties are also offering concessions.

This trend has been seen mostly in markets hit by the double-whammy of negative job growth and new supply, according to executives of one large owner, San Mateo, Calif.-based Essex Property Trust Inc.

“When those two intersect, bad things happen,” said Michael Schall, chief executive of Essex. The company reported weakening market conditions on its quarterly earnings call, sending its shares lower.

During the early years of the recovery from the 2007-09 recession, shares of listed residential real-estate companies far outperformed the real-estate sector. Residential companies were up 40% in 2014, compared with 28% for the broader equity real-estate investment trust market. In 2015, residential companies and the broader REIT market were up 17.1% and 2.8% respectively, according to the National Association of Real Estate Investment Trusts.

This year, as of the end of the September, the gap had narrowed to a 6.9% increase for residential versus 6% for the broader equity REIT sector, the association said.

Some multifamily investors aren’t too worried about rising homeownership, however, because new housing construction continues to lag behind the rate of household formation, even with the surge in rental housing development taken into account. Since 2010, household formation has outpaced construction by about 3.5 million units, according to CoStar Group Inc.

But Wall Street is closely watching demographic trends, particularly marriage rates among millennials—a life change often accompanied by a shift from renting to owning.

Millennials have been getting married later in life, often waiting until their late 20s, according to Mr. Chang, of Marcus & Millichap. Their marriage rate over the next five years will likely play an important role in demand for apartments and houses.

“We’re at the leading edge of transition,” he said.