If you don't want to "slurp" in the whole file at once:

my $name;
my $seq;
while (<DATA>) {
    chomp;
    if (s/^Name:\s*//) {
        $name= $_;
        next;
    }
    if (s/^Nucleotide Sequence:\s*//) {
        $seq= $_;
        while (<DATA>) {
            last if /^s*$/;
            chomp;
            $seq.= $_;
        }
        print "$name\n$seq\n\n";
    }
}

__DATA__
GeneID: 1002
Name: cadherin 4, type 1, R-cadherin (retinal)
Chromo: 20
Cytoband: 20q13.3
Nucleotide Sequence: atgaccgcgggcgccggcgtgctccttctgctgctctcgctctccggc
acagcgagactggagatatcgtcacagtggcggctggcctggaccgagagaaagttcagcagtacacag
cagcttgcgcatcctgtacctggaggccgggatgtatgacgtccccatcatcgtcacagactctggaaa

GeneID: 10077
Name: tetraspanin 32
Chromo: 11
Cytoband: 11p15.5
Nucleotide Sequence: atggggccttggagtcgagtcagggttgccaaatgccagatgctggtc

GeneID: 10078
Name: tumor suppressing subtransferable candidate 4
Chromo: 11
Cytoband: 11p15.5
Nucleotide Sequence: atggctgaggcaggaacaggtgagccgtcccccagcgtggagggcgaa
[download]