Card companies are less likely to give out big limits automatically on application these days, the best thing to do is go for the card you want, build up a balance, pay off at least the minimum for a couple of months then ring up and ask for a credit limit increase - by then they will have built up a risk profile of you and you will have proved to them you can handle the debt.
You are also less likely to get automatic credit limit increases, many lenders are looking to reduce their risk profiles so are putting up APRs and reducing credit limits and focusing only on the customers they make the most money out of.
Cluttered - I doubt it's a problem with your credit rating, it will just be Tesco operating a different risk strategy to other lenders.
Ilikeyoursleeves - I would only apply for one credit card at a time and leave a gap of say 6 months before you get another one, each time you apply for credit, it leaves a footprint on your credit reference file. So to the next lender that checks your credit file it flags you are searching for a lot of credit in a short space of time, which, fairly or not, raises the possibility of you being in financial trouble, or even fraud taking place.