I believe that his "do not be anxious about anything" in Philippians 4:6 describes a different kind of worry than his concern in 2 Corinthians 11:28 (and his fear in 1 Thesalonians 3:4-5). Three key factors lead me to this conclusion.
First, because the Bible is true and trustworthy, I do not believe that we are looking at any kind of inconsistency or sin in Paul. This is not because Paul was perfect, but if he was in the wrong in these instances, the Holy Spirit would have let us in on his sin. His concern was a matter of his human weakness, not a result of a failure on his part to trust God. He couldn't see what was happening in the churches and he wanted assurance that all was well with their faith.
Next, I think the context in Philippians 4 indicates that Paul's prohibition on worry relates to individual joy and contentment in the Lord. Paul learned not to worry about himself and he instructed his readers to pray and not worry about themselves. On the other hand, his concern for others indicates deep love, a pastoral burden that really convicts me in regard to my care for the people in my church.
The last consideration is that the worry of Philippians 4 is probably a non-productive worry, a worry that you can't do anything about (I don't have any good reason for this, it's a hunch based on what I read regarding the other kind of anxiety). This kind of worry is always useless.
Paul's pastoral concern is a kind of anxiety that was justified. The churches that he planted were in danger - danger from false teachers, danger from temptation to sin, danger from overt persecution. These were not imagined problems, they were very present dangers for these churches and they sat heavily on Paul's mind and heart.
More than that, Paul could - and did - do something about these concerns. He prayed, he sent associates like Timothy to places like Thessalonica, he visited these churches himself when he could and, of course, he wrote letters.
So what's the application to us?
When you're anxious, pray. But, when you're anxious, ask the Lord if He is prompting you to do something about that anxiety. If you find that the circumstances that cause you anxiety are beyond your control, pray for contentment. Trust God in the situation. If, however, there is something you can do about someone you are worried about, then do it. It may be a phone call, a visit, a letter, a gift of a useful book, practical help or whatever you come up with.
It could be that that nagging worry is a prompt from God for you, a call to show his love to someone who needs your help and prayer.
Not all absence of worry is a cause for self-congratulation. If I do not have anxiety about other people - good worry, you might say - I may be guilty of not caring enough about people and their circumstances. The Apostle Paul's pastoral care extended to having a measure of productive, prayerful anxiety.
Don't worry: pray, love and serve.