You can get a subsection of the VCF or BAM files using the Ensembl Data Slicer tool. This tool gives you a web interface requesting the URL of any VCF file and the genomic location you wish to get a sub-slice for. This tool also works for BAM files. This tool also allows you to filter the file for particular individuals or populations if you also provide a panel file.
You can also subset VCFs using tabix on the command line, e.g.
tabix -h ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/release/20100804/ALL.2of4intersection.20100804.genotypes.vcf.gz 2:39967768-39967768
Specifications for the VCF format, and a C++ and Perl tool set for VCF files can be found at vcftools on sourceforge
Please note that all our VCF files using straight intergers and X/Y for their chromosome names in the Ensembl style rather than using chr1 in the UCSC style. If you request a subsection of a vcf file using a chromosome name in the style chrN as shown below it will not work.
tabix -h ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/release/20100804/ALL.2of4intersection.20100804.genotypes.vcf.gz chr2:39967768-39967768
You can subset alignment files with samtools on the command line, e.g.
samtools view -h ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/phase1/data/HG00154/alignment/HG00154.mapped.ILLUMINA.bwa.GBR.low_coverage.20101123.bam 17:7512445-7513455
Samtools supports streaming files and piping commands together both using local and remote files. You can get more help with samtools from the samtools help mailing list