lakris@ubuntu:~/projekt/scripts$ cat projekt.txt 
Begin Object1
txt1
end
;
Begin Object2
txt2
end
;
Begin Object3
txt3
end
;
Begin Object4
txt4
end
;
lakris@ubuntu:~/projekt/scripts$ cat splitit.sh 
#!/bin/bash
cnt=0
while read line;do
  [[ "$line" =~ "Begin" ]] && cnt=$(($cnt+1))
  echo $line goes into Object$cnt.txt
done < projekt.txt
lakris@ubuntu:~/projekt/scripts$ ./splitit.sh 
Begin Object1 goes into Object1.txt
txt1 goes into Object1.txt
end goes into Object1.txt
; goes into Object1.txt
Begin Object2 goes into Object2.txt
txt2 goes into Object2.txt
end goes into Object2.txt
; goes into Object2.txt
Begin Object3 goes into Object3.txt
txt3 goes into Object3.txt
end goes into Object3.txt
; goes into Object3.txt
Begin Object4 goes into Object4.txt
txt4 goes into Object4.txt
end goes into Object4.txt
; goes into Object4.txt
lakris@ubuntu:~/projekt/scripts$

awk '/^Begin/{close(f);f=$2".txt"}f{print>f}' input