NEW YORK — Target shoppers found out this weekend that when stores make deals to carry merchandise from high-end designers for a limited time, it can be, well, really limiting.

The discounter partnered with the Lilly Pulitzer brand to carry a collection of 250 pieces for a fraction of the price of the Palm Beach designer’s original merchandise. But the offer, which included $38 pink shift dresses and $25 beach towels, was wrought with long lines in stores, quick sellouts online and other problems.

It shows the challenges stores face when they offer limited-time collections. These lines typically consist of cheaper versions of designer pieces and are sold for a short period of time. They generate buzz from aspirational buyers who want to don upscale brands as well as avid wearers of the labels themselves. But the high demand can be a double-edged sword: Often, customers encounter picked-over merchandise and website snafus.

Target, which pioneered these partnerships in the 1990s and has been followed by rivals like H&M, Gap and Kohl’s, started selling the Lilly Pulitzer collection Sunday online at about 4 a.m. and at stores at 8 a.m. Demand was so heavy that Target took the site offline for 20 minutes, which caused angry chatter on social media. Ultimately, the items sold out online within a few hours and at many of the 1,800 stores within a half-hour.