Sadly, it is true that many people donate such junk with the attitude "They should feel lucky to get this. After all, they have nothing right now." I've seen the junk pass through church rummage sales and someone just has to take the time to surreptitiously throw it in the garbage to avoid offending the donor.
Wander through most any church building and consider the mismatched furniture you will likely find there. Consider how uncomfortable it usually is. Consider that this furniture was probably donated to the church because someone else was getting new furniture and didn't want this furniture anymore however "the Church should feel lucky to get this."
I expect this was the same attitude which inspired people from around the world to donate evening dresses, thong underwear and stiletto high heel shoes to the Tsunami relief effort. In the same vein, roads in southeast Asia were clogged with trucks hauling useless supplies -- entire containers of ground black pepper made their way to Sri Lanka. The country, which experienced a bumper crop of rice that year, was forced to place a tax on rice that was shipped to the country -- without that taxation the local rice economy would have collapsed.
One lady at Church came up to me and asked if I thought a local mission congregation of our denomination would want our old worship banners that were mouldering in the basement of the church. My immediate and loud response was, "No!" She asked, "Why wouldn't they want them? They don't have any banners at that church." I asked her, "Why are we getting rid of them?" She responded, "Because they are old and falling apart and they don't look good in the Church." My final response was, "So -- if we don't want them because they are ugly, what makes you think they would want them either?"
In response to Hurrican Katrina and the quest for new only donations -- surely everyone who wants to donate some clothing can afford to go to Wal-Mart and buy a package of new underwear or a new T-shirt or a new pair of pants and donate those to the relief effort. By purchasing such items at Wal-Mart you'll help support local economies by employing Wal-mart employees and you'll help employ workers in sweatshops around the world who produce products companies such as Levi Strauss import to sell at Wal-Mart.