Is it possible to use grep to match only lines with numbers in a pre-specified range?
For instance I want to list all lines with numbers in the range [1024, 2048] of a log that contain the word 'error'. 
I would like to keep the '-n' functionality i.e. have the number of the matched line in the file.
sed -n '1024,2048{/error/{=;p}}' | paste - -
Here /error/ is a pattern to match and = prints the line number.
Use sed first:
sed -ne '1024,2048p' | grep ...
-n says don't print lines, 'x,y,p' says print lines x-y inclusive (overrides the -n)
Awk is a good tool for the job: 
$ awk 'NR>=1024 && NR<=2048 && /error/ {print NR,$0}' file
In awk the variable NR contains the current line number and $0 contains the line itself.
The benefits with using awk is that you can easily change the output to display however you want it. For instance to separate the line number from line with a colon followed by a TAB:
$ awk 'NR>=1024 && NR<=2048 && /error/ {print NR,$0}' OFS=':\t' file
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