Chapter 5:3,4 7,8, and 16 are an exhortation to make a commitment to relieving the needs of a widowed family member (mother, grandmother, aunt, etc.) as a way of properly honoring her (them) and as an opportunity to repay the loved one for her investment in the believer, and as a way of demonstrating sincere faith toward God.
A case can be made that such care may well require financial resources in excess of current income, and that accumulation in anticipation of this need is justified. I would agree—more importantly, the Lord would agree (II Corinthians 8:14) that reasonable savings are a part of stewardship, but let's keep in mind that unreasonable (excessive) savings bring condemnation (Luke 12:16-21; Proverbs 1:32)—the very condemnation we have been unnecessarily concerned about as we have misapplied I Timothy 5:8.
It seems clear to me that acknowledging the Lord to be our provider, and clearly demonstrating to Him that we have yielded that role to its rightful owner through a conscious effort to not overaccumulate, is a very important part of our testimony of "love out of a pure heart, and of a good conscience, and of faith unfeigned."
A by-product of that testimony will be loved ones cared for; needs met during unemployment, disability and illness relieved; missionaries sent; churches planted; the Great Commission fulfilled—all to the glory of God. I wonder if you have had the same experience with this portion of Scripture that I have had. Do you find it to be the first thing that comes to your mind when you ask yourself, "How can we justify keeping so much for ourselves when there are so many unmet needs around us?" If you are like me, this one reference to my responsibility to provide for my family has enabled me to override the scores of references in Scripture where God reserves to Himself the role of provider. I am afraid I Timothy 5:8 has become, for many in the church, an excuse for excessive accumulation.
In context, the verse has absolutely nothing to do with accumulation—it has everything to do with the Christian taking responsibility for needy widow(s) in his/her family! Paul begins the letter by warning Timothy to be on guard lest false doctrines destroy godly edifying which comes from love "out of a pure heart, and of a good conscience, and of faith unfeigned (sincere)" (I Timothy 1:5). He gives instruction as to godly living for both men and women in chapter 2: enumerates the qualifications for leaders of the church in chapter 3 (taking pains to point out that the evidence of a proper heart is proper conduct); and in chapter 4, warns again about apostasy and prescribes its antidote—faithful teaching by one whose life is wholly and visibly committed to God (vv. 6,11,13,15,16).